{"id":2779,"date":"2023-05-15T17:49:03","date_gmt":"2023-05-15T17:49:03","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=2779"},"modified":"2024-10-18T20:51:41","modified_gmt":"2024-10-18T20:51:41","slug":"volume-and-surface-area-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/volume-and-surface-area-fresh-take\/","title":{"raw":"Volume and Surface Area: Fresh Take","rendered":"Volume and Surface Area: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Find the volume of a rectangular solid, sphere, cylinder, and cone<\/li>\r\n\t<li>Find the surface area of a rectangular solid, sphere, and cylinder<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Finding the Volume and Surface Area of Rectangular Solids<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>The <strong>volume <\/strong>of a rectangular solid is a measurement that tells us how much space is enclosed or occupied by the solid in three dimensions. It helps us understand the capacity or quantity of material that can fit within the shape. Since volume is a measure of three-dimensional space, the units will be in cubic units.<\/p>\r\n<p>To calculate the volume of a rectangular solid, we multiply its three dimensions: length, width, and height. The formula for finding the volume is [latex]V = L \u00d7 W \u00d7 H[\/latex], where [latex]V[\/latex] represents the volume, [latex]L[\/latex] represents the length, [latex]W[\/latex] represents the width, and [latex]H[\/latex] represents the height. This can also be written as [latex]V = B \u00d7 H[\/latex], where [latex]B[\/latex] is the area of the base - [latex]B = L \u00d7 W[\/latex].<\/p>\r\n<p>The <strong>surface area<\/strong> of a rectangular solid is a measurement that tells us the total area of all the exposed surfaces of the solid. It helps us understand the amount of material needed to cover the shape. Since surface area is a measure of two-dimensional space, the units will be in square units.<\/p>\r\n<p>To calculate the surface area of a rectangular solid, we add up the areas of all its faces. For a rectangular solid, there are six faces: the top, bottom, front, back, left side, and right side. Each face is a rectangle, so we can find the area of each face by multiplying its length and width. Finally, we sum up the areas of all the faces to find the total surface area. The formula for finding the surface area of a rectangular solid is [latex]SA = 2LW + 2LH + 2WH[\/latex], where [latex]SA[\/latex] represents the surface area, [latex]L[\/latex] represents the length, [latex]W[\/latex] represents the width, and [latex]H[\/latex] represents the height of the rectangular solid.<\/p>\r\n<\/div>\r\n<section class=\"textbox example\">A rectangular crate has a length of [latex]30[\/latex] inches, width of [latex]25[\/latex] inches, and height of [latex]20[\/latex] inches. Find its:\r\n\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>volume<\/li>\r\n\t<li>surface area<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"57883\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57883\"]<\/p>\r\n<p>Step 1 is the same for both 1. and 2., so we will show it just once.<\/p>\r\n<table id=\"eip-id1168467275051\" class=\"unnumbered unstyled\" summary=\"The text reads, \">\r\n<tbody>\r\n<tr>\r\n<td>\r\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and<\/p>\r\n<p>label it with the given information.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224141\/CNX_BMath_Figure_09_06_039_img-01.png\" alt=\"A rectangular prism with one side labeled 20, another side labeled 25, and another side labeled 30.\" width=\"173\" height=\"115\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the volume of the crate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>let [latex]V[\/latex] = volume<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula.<\/p>\r\n<p>Substitute.<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]V=LWH[\/latex]<\/p>\r\n<p>[latex]V=30\\cdot 25\\cdot 20[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]V=15,000[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> Double check your math.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The volume is [latex]15,000[\/latex] cubic inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the surface area of the crate<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>let [latex]S[\/latex] = surface area<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula. Substitute.<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]S=2LH+2LW+2WH[\/latex]<\/p>\r\n<p>[latex]S=2\\left(30\\cdot 20\\right)+2\\left(30\\cdot 25\\right)+2\\left(25\\cdot 20\\right)[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]S=3,700[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> Check it yourself!<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The surface area is [latex]3,700[\/latex] square inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Finding the Volume and Surface Area of a Cube<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>A <strong>cube<\/strong> is a three-dimensional shape with equal sides, forming six identical square faces.<\/p>\r\n<p>To find the <strong>volume <\/strong>of a cube, you raise the length of one side to the power of three. The formula for calculating the volume of a cube is [latex]V = s^3[\/latex], where [latex]V[\/latex] represents the volume and [latex]s[\/latex] represents the length of one side.<\/p>\r\n<p>The <strong>surface area<\/strong> of a cube is the sum of the areas of all its faces. Since all the faces are squares with equal sides, you can multiply the length of one side by itself and then multiply by six to account for all the faces. The formula for finding the <strong>surface area<\/strong> of a cube is [latex]SA = 6s^2[\/latex], where [latex]SA[\/latex] represents the surface area and [latex]s[\/latex] represents the length of one side.<\/p>\r\n<\/div>\r\n<section class=\"textbox example\">A notepad cube measures [latex]2[\/latex] inches on each side. Find its:\r\n\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>volume<\/li>\r\n\t<li>surface area<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"57881\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57881\"]<\/p>\r\n<table id=\"eip-id1168467296950\" class=\"unnumbered unstyled\" summary=\"The text reads, \">\r\n<tbody>\r\n<tr>\r\n<td>\r\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and<\/p>\r\n<p>label it with the given information.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224144\/CNX_BMath_Figure_09_06_041_img-01.png\" alt=\"A cube with each side labeled 2\" width=\"141\" height=\"127\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168467188434\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the volume of the cube<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>let <em>V<\/em> = volume<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula.<\/p>\r\n<\/td>\r\n<td>[latex]V={s}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>\r\n<p>[latex]V={2}^{3}[\/latex]<\/p>\r\n<p>[latex]V=8[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 6. <strong>Check:<\/strong> Check that you did the calculations<\/p>\r\n<p>correctly.<\/p>\r\n<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The volume is [latex]8[\/latex] cubic inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168468478236\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the surface area of the cube<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>let <em>S<\/em> = surface area<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula.<\/p>\r\n<\/td>\r\n<td>[latex]S=6{s}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>\r\n<p>[latex]S=6\\cdot {2}^{2}[\/latex]<\/p>\r\n<p>[latex]S=24[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> The check is left to you.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The surface area is [latex]24[\/latex] square inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Finding the Volume and Surface Area of a Sphere<\/h2>\r\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<br \/>\r\n<\/strong><br \/>\r\nA sphere is a three-dimensional shape with all points equidistant from its center. To find the <strong>volume <\/strong>of a sphere, you use the formula [latex]V = (4\/3)\u03c0r^3[\/latex], where [latex]V[\/latex] represents the volume and [latex]r[\/latex] represents the radius of the sphere. The radius is the distance from the center of the sphere to any point on its surface. The <strong>surface area<\/strong> of a sphere is the total area covered by its outer surface. You can calculate it using the formula [latex]SA = 4\u03c0r^2[\/latex], where [latex]SA[\/latex] represents the surface area and [latex]r[\/latex] represents the radius of the sphere.<\/div>\r\n<section class=\"textbox example\">A globe of Earth is in the shape of a sphere with radius [latex]14[\/latex] centimeters. Find its:\r\n\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>volume<\/li>\r\n\t<li>surface area<\/li>\r\n<\/ol>\r\n<p>Round the answer to the nearest hundredth.<\/p>\r\n<p>[reveal-answer q=\"57882\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57882\"]<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>\r\n<p>Step 1. <strong>Read<\/strong> the problem. Draw a figure with the<\/p>\r\n<p>given information and label it.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224147\/CNX_BMath_Figure_09_06_043_img-01.png\" alt=\"A globe of the Earth with radius 14\" width=\"150\" height=\"151\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the volume of the sphere<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>let [latex]V[\/latex] = volume<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula. Substitute. (Use 3.14 for [latex]\\pi [\/latex] )<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]V=\\Large\\frac{4}{3}\\normalsize\\pi {r}^{3}[\/latex]<\/p>\r\n<p>[latex]V\\approx \\Large\\frac{4}{3}\\normalsize\\left(3.14\\right){14}^{3}[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex]V\\approx 11,488.21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your calculations.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The volume is approximately [latex]11,488.21[\/latex] cubic inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the surface area of the sphere<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>let <em>S<\/em> = surface area<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula. Substitute. (Use 3.14 for [latex]\\pi [\/latex] )<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]S=4\\pi {r}^{2}[\/latex]<\/p>\r\n<p>[latex]S\\approx 4\\left(3.14\\right){14}^{2}[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex]S\\approx 2461.76[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your calculations.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The surface area is approximately [latex]2461.76[\/latex] square inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>The following video shows an example of how to find the surface area of a sphere.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/OLlEbk1xGm4\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Find+the+Surface+Area+of+a+Sphere.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFind the Surface Area of a Sphere\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>In the next video we show an example of how to find the volume of a sphere given it's diameter.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/_kejcXbRjGY\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Volume+of+a+Sphere.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Volume of a Sphere\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Finding the Volume and Surface Area of a Cylinder<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>A <strong>cylinder <\/strong>is a three-dimensional shape with two circular bases and a curved surface connecting the bases.<\/p>\r\n<p>To find the <strong>volume<\/strong> of a cylinder, you multiply the area of the base (which is a circle) by the height of the cylinder. The formula for calculating the volume of a cylinder is [latex]V = \u03c0r^2h[\/latex], where [latex]V[\/latex] represents the volume, [latex]r[\/latex] represents the radius of the base, and [latex]h[\/latex] represents the height of the cylinder.<\/p>\r\n<p>The surface area of a cylinder is the sum of the areas of its curved surface and the two circular bases. The formula for finding the surface area of a cylinder is [latex]SA = 2\u03c0rh + 2\u03c0r^2[\/latex], where [latex]SA[\/latex] represents the surface area, [latex]r[\/latex] represents the radius of the base, and [latex]h[\/latex] represents the height of the cylinder.<\/p>\r\n<\/div>\r\n<section class=\"textbox example\">The radius of the base of a soda can is [latex]4[\/latex] centimeters and the height is [latex]13[\/latex] centimeters. Assume the can is shaped exactly like a cylinder.<br \/>\r\nFind its:\r\n\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>volume<\/li>\r\n\t<li>surface area<\/li>\r\n<\/ol>\r\n<p>Round the answer to the nearest hundredth.<\/p>\r\n<p>[reveal-answer q=\"57889\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"57889\"]<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>\r\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and <br \/>\r\nlabel it with the given information.<\/p>\r\n<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224157\/CNX_BMath_Figure_09_06_047_img-01.png\" alt=\"A can with height 13 and radius of the base 4.\" width=\"116\" height=\"147\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the volume of the cylinder<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>let <em>V<\/em> = volume<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula. Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi [\/latex] )<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]V=\\pi {r}^{2}h[\/latex]<\/p>\r\n<p>[latex]V\\approx \\left(3.14\\right){4}^{2}\\cdot 13[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex]V\\approx 653.12[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The volume is approximately [latex]653.12[\/latex] cubic centimeters.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t<li>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the surface area of the cylinder<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>let <em>S<\/em> = surface area<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 4. <strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula. Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi [\/latex] )<\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]S=2\\pi {r}^{2}+2\\pi rh[\/latex]<\/p>\r\n<p>[latex]S\\approx 2\\left(3.14\\right){4}^{2}+2\\left(3.14\\right)\\left(4\\right)13[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex]S\\approx 427.04[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your calculations.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The surface area is approximately [latex]427.04[\/latex] square centimeters.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>The following video shows an example of how to find the volume of a cylinder.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/oDgfx-Kztrk\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Volume+of+a+Cylinder.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Volume of a Cylinder\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<p>In the next example video we show how to find the surface area of a cylinder.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/CN_7ZxmixXY\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Find+the+surface+Area+of+a+Cylinder.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFind the surface Area of a Cylinder\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Finding the Volume of a Cone<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>A <strong>cone <\/strong>is a three-dimensional shape with a circular base and a pointed top.<\/p>\r\n<p>To find the <strong>volume <\/strong>of a cone, you multiply the area of the base (which is a circle) by the height of the cone and divide by [latex]3[\/latex]. The formula for calculating the volume of a cone is [latex]V = \\frac{1}{3}\u03c0r^2h[\/latex], where [latex]V[\/latex] represents the volume, [latex]r[\/latex] represents the radius of the base, and [latex]h[\/latex] represents the height of the cone.<\/p>\r\n<\/div>\r\n<section class=\"textbox example\">Marty\u2019s favorite pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is [latex]8[\/latex] inches tall and [latex]5[\/latex] inches in diameter? Round the answer to the nearest hundredth.[reveal-answer q=\"499965\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"499965\"]\r\n\r\n\r\n<table id=\"eip-id1168469874277\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information. Notice here that the base is the circle at the top of the cone.<\/td>\r\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224204\/CNX_BMath_Figure_09_06_049_img-01.png\" alt=\"A cone with height 8 and radius of the base 5.\" width=\"121\" height=\"145\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the volume of the cone<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>let <em>V<\/em> = volume<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula. Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi [\/latex] , and notice that we were given the distance across the circle, which is its diameter. The radius is [latex]2.5[\/latex] inches.)<\/td>\r\n<td>\r\n<p>[latex]V=\\Large\\frac{1}{3}\\normalsize\\pi {r}^{2}h[\/latex]<\/p>\r\n<p>[latex]V\\approx \\Large\\frac{1}{3}\\normalsize 3.14{\\left(2.5\\right)}^{2}\\left(8\\right)[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex]V\\approx 52.33[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your calculations.<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The volume of the wrap is approximately [latex]52.33[\/latex] cubic inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>In the following video we provide another example of how to find the volume of a cone.<\/p>\r\n<section class=\"textbox watchIt\"><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/7Y0ZMnCcVGs\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Determine+Volume+of+a+Cone.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Determine Volume of a Cone\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Find the volume of a rectangular solid, sphere, cylinder, and cone<\/li>\n<li>Find the surface area of a rectangular solid, sphere, and cylinder<\/li>\n<\/ul>\n<\/section>\n<h2>Finding the Volume and Surface Area of Rectangular Solids<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>The <strong>volume <\/strong>of a rectangular solid is a measurement that tells us how much space is enclosed or occupied by the solid in three dimensions. It helps us understand the capacity or quantity of material that can fit within the shape. Since volume is a measure of three-dimensional space, the units will be in cubic units.<\/p>\n<p>To calculate the volume of a rectangular solid, we multiply its three dimensions: length, width, and height. The formula for finding the volume is [latex]V = L \u00d7 W \u00d7 H[\/latex], where [latex]V[\/latex] represents the volume, [latex]L[\/latex] represents the length, [latex]W[\/latex] represents the width, and [latex]H[\/latex] represents the height. This can also be written as [latex]V = B \u00d7 H[\/latex], where [latex]B[\/latex] is the area of the base &#8211; [latex]B = L \u00d7 W[\/latex].<\/p>\n<p>The <strong>surface area<\/strong> of a rectangular solid is a measurement that tells us the total area of all the exposed surfaces of the solid. It helps us understand the amount of material needed to cover the shape. Since surface area is a measure of two-dimensional space, the units will be in square units.<\/p>\n<p>To calculate the surface area of a rectangular solid, we add up the areas of all its faces. For a rectangular solid, there are six faces: the top, bottom, front, back, left side, and right side. Each face is a rectangle, so we can find the area of each face by multiplying its length and width. Finally, we sum up the areas of all the faces to find the total surface area. The formula for finding the surface area of a rectangular solid is [latex]SA = 2LW + 2LH + 2WH[\/latex], where [latex]SA[\/latex] represents the surface area, [latex]L[\/latex] represents the length, [latex]W[\/latex] represents the width, and [latex]H[\/latex] represents the height of the rectangular solid.<\/p>\n<\/div>\n<section class=\"textbox example\">A rectangular crate has a length of [latex]30[\/latex] inches, width of [latex]25[\/latex] inches, and height of [latex]20[\/latex] inches. Find its:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>volume<\/li>\n<li>surface area<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57883\">Show Solution<\/button><\/p>\n<div id=\"q57883\" class=\"hidden-answer\" style=\"display: none\">\n<p>Step 1 is the same for both 1. and 2., so we will show it just once.<\/p>\n<table id=\"eip-id1168467275051\" class=\"unnumbered unstyled\" summary=\"The text reads,\">\n<tbody>\n<tr>\n<td>\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and<\/p>\n<p>label it with the given information.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224141\/CNX_BMath_Figure_09_06_039_img-01.png\" alt=\"A rectangular prism with one side labeled 20, another side labeled 25, and another side labeled 30.\" width=\"173\" height=\"115\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol style=\"list-style-type: decimal;\">\n<li>\n<table>\n<tbody>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the volume of the crate<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>let [latex]V[\/latex] = volume<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute.<\/p>\n<\/td>\n<td>\n[latex]V=LWH[\/latex]<br \/>\n[latex]V=30\\cdot 25\\cdot 20[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]V=15,000[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> Double check your math.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The volume is [latex]15,000[\/latex] cubic inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the surface area of the crate<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>let [latex]S[\/latex] = surface area<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula. Substitute.<\/p>\n<\/td>\n<td>\n[latex]S=2LH+2LW+2WH[\/latex]<br \/>\n[latex]S=2\\left(30\\cdot 20\\right)+2\\left(30\\cdot 25\\right)+2\\left(25\\cdot 20\\right)[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]S=3,700[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> Check it yourself!<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The surface area is [latex]3,700[\/latex] square inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<h2>Finding the Volume and Surface Area of a Cube<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>A <strong>cube<\/strong> is a three-dimensional shape with equal sides, forming six identical square faces.<\/p>\n<p>To find the <strong>volume <\/strong>of a cube, you raise the length of one side to the power of three. The formula for calculating the volume of a cube is [latex]V = s^3[\/latex], where [latex]V[\/latex] represents the volume and [latex]s[\/latex] represents the length of one side.<\/p>\n<p>The <strong>surface area<\/strong> of a cube is the sum of the areas of all its faces. Since all the faces are squares with equal sides, you can multiply the length of one side by itself and then multiply by six to account for all the faces. The formula for finding the <strong>surface area<\/strong> of a cube is [latex]SA = 6s^2[\/latex], where [latex]SA[\/latex] represents the surface area and [latex]s[\/latex] represents the length of one side.<\/p>\n<\/div>\n<section class=\"textbox example\">A notepad cube measures [latex]2[\/latex] inches on each side. Find its:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>volume<\/li>\n<li>surface area<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57881\">Show Solution<\/button><\/p>\n<div id=\"q57881\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168467296950\" class=\"unnumbered unstyled\" summary=\"The text reads,\">\n<tbody>\n<tr>\n<td>\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and<\/p>\n<p>label it with the given information.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224144\/CNX_BMath_Figure_09_06_041_img-01.png\" alt=\"A cube with each side labeled 2\" width=\"141\" height=\"127\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168467188434\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the volume of the cube<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>let <em>V<\/em> = volume<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<\/td>\n<td>[latex]V={s}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>\n[latex]V={2}^{3}[\/latex]<br \/>\n[latex]V=8[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 6. <strong>Check:<\/strong> Check that you did the calculations<\/p>\n<p>correctly.<\/p>\n<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The volume is [latex]8[\/latex] cubic inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168468478236\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the surface area of the cube<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>let <em>S<\/em> = surface area<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<\/td>\n<td>[latex]S=6{s}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>\n[latex]S=6\\cdot {2}^{2}[\/latex]<br \/>\n[latex]S=24[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> The check is left to you.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The surface area is [latex]24[\/latex] square inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<h2>Finding the Volume and Surface Area of a Sphere<\/h2>\n<div class=\"textbox shaded\"><strong>The Main Idea\u00a0<br \/>\n<\/strong><br \/>\nA sphere is a three-dimensional shape with all points equidistant from its center. To find the <strong>volume <\/strong>of a sphere, you use the formula [latex]V = (4\/3)\u03c0r^3[\/latex], where [latex]V[\/latex] represents the volume and [latex]r[\/latex] represents the radius of the sphere. The radius is the distance from the center of the sphere to any point on its surface. The <strong>surface area<\/strong> of a sphere is the total area covered by its outer surface. You can calculate it using the formula [latex]SA = 4\u03c0r^2[\/latex], where [latex]SA[\/latex] represents the surface area and [latex]r[\/latex] represents the radius of the sphere.<\/div>\n<section class=\"textbox example\">A globe of Earth is in the shape of a sphere with radius [latex]14[\/latex] centimeters. Find its:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>volume<\/li>\n<li>surface area<\/li>\n<\/ol>\n<p>Round the answer to the nearest hundredth.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57882\">Show Solution<\/button><\/p>\n<div id=\"q57882\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td>\n<p>Step 1. <strong>Read<\/strong> the problem. Draw a figure with the<\/p>\n<p>given information and label it.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224147\/CNX_BMath_Figure_09_06_043_img-01.png\" alt=\"A globe of the Earth with radius 14\" width=\"150\" height=\"151\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol style=\"list-style-type: decimal;\">\n<li>\n<table>\n<tbody>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the volume of the sphere<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>let [latex]V[\/latex] = volume<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula. Substitute. (Use 3.14 for [latex]\\pi[\/latex] )<\/p>\n<\/td>\n<td>\n[latex]V=\\Large\\frac{4}{3}\\normalsize\\pi {r}^{3}[\/latex]<br \/>\n[latex]V\\approx \\Large\\frac{4}{3}\\normalsize\\left(3.14\\right){14}^{3}[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]V\\approx 11,488.21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your calculations.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The volume is approximately [latex]11,488.21[\/latex] cubic inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the surface area of the sphere<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>let <em>S<\/em> = surface area<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula. Substitute. (Use 3.14 for [latex]\\pi[\/latex] )<\/p>\n<\/td>\n<td>\n[latex]S=4\\pi {r}^{2}[\/latex]<br \/>\n[latex]S\\approx 4\\left(3.14\\right){14}^{2}[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]S\\approx 2461.76[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your calculations.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The surface area is approximately [latex]2461.76[\/latex] square inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>The following video shows an example of how to find the surface area of a sphere.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/OLlEbk1xGm4\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Find+the+Surface+Area+of+a+Sphere.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFind the Surface Area of a Sphere\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>In the next video we show an example of how to find the volume of a sphere given it&#8217;s diameter.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/_kejcXbRjGY\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Volume+of+a+Sphere.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Volume of a Sphere\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Finding the Volume and Surface Area of a Cylinder<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>A <strong>cylinder <\/strong>is a three-dimensional shape with two circular bases and a curved surface connecting the bases.<\/p>\n<p>To find the <strong>volume<\/strong> of a cylinder, you multiply the area of the base (which is a circle) by the height of the cylinder. The formula for calculating the volume of a cylinder is [latex]V = \u03c0r^2h[\/latex], where [latex]V[\/latex] represents the volume, [latex]r[\/latex] represents the radius of the base, and [latex]h[\/latex] represents the height of the cylinder.<\/p>\n<p>The surface area of a cylinder is the sum of the areas of its curved surface and the two circular bases. The formula for finding the surface area of a cylinder is [latex]SA = 2\u03c0rh + 2\u03c0r^2[\/latex], where [latex]SA[\/latex] represents the surface area, [latex]r[\/latex] represents the radius of the base, and [latex]h[\/latex] represents the height of the cylinder.<\/p>\n<\/div>\n<section class=\"textbox example\">The radius of the base of a soda can is [latex]4[\/latex] centimeters and the height is [latex]13[\/latex] centimeters. Assume the can is shaped exactly like a cylinder.<br \/>\nFind its:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>volume<\/li>\n<li>surface area<\/li>\n<\/ol>\n<p>Round the answer to the nearest hundredth.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q57889\">Show Solution<\/button><\/p>\n<div id=\"q57889\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td>\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and <br \/>\nlabel it with the given information.<\/p>\n<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224157\/CNX_BMath_Figure_09_06_047_img-01.png\" alt=\"A can with height 13 and radius of the base 4.\" width=\"116\" height=\"147\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol style=\"list-style-type: decimal;\">\n<li>\n<table>\n<tbody>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the volume of the cylinder<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>let <em>V<\/em> = volume<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula. Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi[\/latex] )<\/p>\n<\/td>\n<td>\n[latex]V=\\pi {r}^{2}h[\/latex]<br \/>\n[latex]V\\approx \\left(3.14\\right){4}^{2}\\cdot 13[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]V\\approx 653.12[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The volume is approximately [latex]653.12[\/latex] cubic centimeters.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the surface area of the cylinder<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>let <em>S<\/em> = surface area<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula. Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi[\/latex] )<\/p>\n<\/td>\n<td>\n[latex]S=2\\pi {r}^{2}+2\\pi rh[\/latex]<br \/>\n[latex]S\\approx 2\\left(3.14\\right){4}^{2}+2\\left(3.14\\right)\\left(4\\right)13[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]S\\approx 427.04[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your calculations.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The surface area is approximately [latex]427.04[\/latex] square centimeters.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>The following video shows an example of how to find the volume of a cylinder.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/oDgfx-Kztrk\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Volume+of+a+Cylinder.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Volume of a Cylinder\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<p>In the next example video we show how to find the surface area of a cylinder.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/CN_7ZxmixXY\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Find+the+surface+Area+of+a+Cylinder.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFind the surface Area of a Cylinder\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Finding the Volume of a Cone<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>A <strong>cone <\/strong>is a three-dimensional shape with a circular base and a pointed top.<\/p>\n<p>To find the <strong>volume <\/strong>of a cone, you multiply the area of the base (which is a circle) by the height of the cone and divide by [latex]3[\/latex]. The formula for calculating the volume of a cone is [latex]V = \\frac{1}{3}\u03c0r^2h[\/latex], where [latex]V[\/latex] represents the volume, [latex]r[\/latex] represents the radius of the base, and [latex]h[\/latex] represents the height of the cone.<\/p>\n<\/div>\n<section class=\"textbox example\">Marty\u2019s favorite pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is [latex]8[\/latex] inches tall and [latex]5[\/latex] inches in diameter? Round the answer to the nearest hundredth.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q499965\">Show Solution<\/button><\/p>\n<div id=\"q499965\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168469874277\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information. Notice here that the base is the circle at the top of the cone.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224204\/CNX_BMath_Figure_09_06_049_img-01.png\" alt=\"A cone with height 8 and radius of the base 5.\" width=\"121\" height=\"145\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the volume of the cone<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>let <em>V<\/em> = volume<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula. Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi[\/latex] , and notice that we were given the distance across the circle, which is its diameter. The radius is [latex]2.5[\/latex] inches.)<\/td>\n<td>\n[latex]V=\\Large\\frac{1}{3}\\normalsize\\pi {r}^{2}h[\/latex]<br \/>\n[latex]V\\approx \\Large\\frac{1}{3}\\normalsize 3.14{\\left(2.5\\right)}^{2}\\left(8\\right)[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]V\\approx 52.33[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> We leave it to you to check your calculations.<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The volume of the wrap is approximately [latex]52.33[\/latex] cubic inches.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<p>In the following video we provide another example of how to find the volume of a cone.<\/p>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/7Y0ZMnCcVGs\" width=\"560\" height=\"315\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Ex_+Determine+Volume+of+a+Cone.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Determine Volume of a Cone\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":32,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":71,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2779"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":39,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2779\/revisions"}],"predecessor-version":[{"id":15340,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2779\/revisions\/15340"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/71"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2779\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=2779"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=2779"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=2779"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=2779"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}