{"id":454,"date":"2025-02-13T19:45:07","date_gmt":"2025-02-13T19:45:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/physical-applications-of-integration-background-youll-need-2\/"},"modified":"2025-02-13T19:45:07","modified_gmt":"2025-02-13T19:45:07","slug":"physical-applications-of-integration-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/physical-applications-of-integration-background-youll-need-2\/","title":{"raw":"Physical Applications of Integration: Background You'll Need 2","rendered":"Physical Applications of Integration: Background You&#8217;ll Need 2"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Use geometric formulas to find the volume, area, and perimeter of shapes in real-life problems<\/li>\n<\/ul>\n<\/section>\n<p>Understanding how to apply geometric formulas is essential for solving practical problems you will encounter in calculus and everyday life. These skills are particularly useful in various physical applications such as determining the mass of objects, calculating work done by variable forces, and finding the hydrostatic force against submerged plates.<\/p>\n<h2>Appling Geometric Formulas to Solve for Volume, Area, and Perimeter<\/h2>\n<p>To effectively apply these geometric formulas, it's essential to understand the components of each formula and how they relate to the shapes involved. By mastering these basic principles, you will be better equipped to solve a variety of practical problems in both academic and real-world contexts.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>essential geometric formulas<\/h3>\n<p>To solve practical problems involving geometry, remember the key formulas:<\/p>\n<ul>\n\t<li><strong>Volume<\/strong>:\n\n<ul>\n\t<li>Rectangular Prism: [latex]V = l \\times w \\times h [\/latex]&nbsp;<\/li>\n\t<li>Cylinder:&nbsp; [latex]V = \\pi r^2 h[\/latex]&nbsp;<\/li>\n\t<li>Sphere: V = [latex]\\frac{4}{3} \\pi r^3[\/latex]&nbsp;<\/li>\n<\/ul>\n<\/li>\n\t<li><strong>Area<\/strong>:\n\n<ul>\n\t<li>Rectangle: [latex]A = l \\times w[\/latex]<\/li>\n\t<li>Triangle: [latex]A = \\frac{1}{2} b \\times h[\/latex]&nbsp;<\/li>\n\t<li>Circle: [latex]A = \\pi r^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n\t<li><strong>Perimeter<\/strong>:\n\n<ul>\n\t<li>Rectangle: [latex]P = 2l + 2w[\/latex]<\/li>\n\t<li>Triangle: [latex]P = a + b + c[\/latex]<\/li>\n\t<li>Circle (Circumference): [latex]C = 2\\pi r[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>Here, [latex]l[\/latex] stands for length, [latex]w[\/latex] for width, [latex]h[\/latex] for height, [latex]r[\/latex] for radius, [latex]b[\/latex] for base (in area of a triangle formula), [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] for the sides of a triangle (in perimeter of a triangle formula), and [latex]\\pi[\/latex] is the constant Pi (approximately [latex]3.14159[\/latex]).<\/p>\n<p>&nbsp;<\/p>\n<\/section>\n<section class=\"textbox questionHelp\"><strong>How to: Solve Volume, Area, and Perimeter Problems<\/strong>\n<ol style=\"list-style-type: lower-alpha;\">\n\t<li><strong>Identify the shape:<\/strong> Determine whether you are working with a rectangle, triangle, circle, cylinder, etc.<\/li>\n\t<li><strong>Choose the appropriate formula:<\/strong> Select the formula that corresponds to the shape and the measurement you need to find (volume, area, or perimeter).<\/li>\n\t<li><strong>Substitute the given values:<\/strong> Plug in the values provided in the problem into the formula.<\/li>\n\t<li><strong>Solve the equation:<\/strong> Perform the calculations to find the answer.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox proTip\">\n<p>When working with geometry formulas, we recommend using the following problem-solving strategy when solving.<\/p>\n<p><strong>Problem-Solving Strategy for Geometry Applications<\/strong><\/p>\n<ol id=\"eip-id1170325410595\" class=\"stepwise\">\n\t<li><strong>Read<\/strong> the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.<\/li>\n\t<li><strong>Identify<\/strong> what you are looking for.<\/li>\n\t<li><strong>Name<\/strong> what you are looking for and choose a variable to represent it.<\/li>\n\t<li><strong>Translate<\/strong> into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\n\t<li><strong>Solve<\/strong> the equation using good algebra techniques.<\/li>\n\t<li><strong>Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n\t<li><strong>Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>The length of a rectangular playground is [latex]32[\/latex] meters and the width is [latex]20[\/latex] meters. Find the<\/p>\n<ol style=\"list-style-type: decimal;\">\n\t<li>Perimeter of the rectangular playground<\/li>\n\t<li>Area of the rectangular playground<\/li>\n<\/ol>\n<p>[reveal-answer q=\"172561\"]Show Solution[\/reveal-answer] [hidden-answer a=\"172561\"]<\/p>\n<ol>\n\t<li>\n<table>\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223840\/CNX_BMath_Figure_09_04_067_img_MW-01.png\" alt=\"A rectangle with the top and bottom labeled 32 m and the sides labeled 20 m\" width=\"303\" height=\"174\"><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the perimeter of a rectangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]P[\/latex] = the perimeter<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula. Substitute.<\/td>\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223841\/CNX_BMath_Figure_09_04_067_img_MW-02.png\" alt=\"The formula P = 2L + 2W. The formula is then written again with 32 substituted in for L and 20 substituted in for W\" width=\"524\" height=\"100\"><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]P=64+40[\/latex] [latex]P=104[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong><\/td>\n<td>\n<p>[latex]p\\stackrel{?}{=}104[\/latex]<\/p>\n<p>[latex]20+32+20+32\\stackrel{?}{=}104[\/latex]<\/p>\n<p>[latex]104=104\\checkmark[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The perimeter of the rectangle is [latex]104[\/latex] meters.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n\t<li>\n<table>\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223845\/CNX_BMath_Figure_09_04_068_img_MW-01.png\" alt=\"A rectangle with the top and bottom labeled 32 m and the sides labeled 20 m\" width=\"310\" height=\"176\"><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the area of a rectangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>A<\/em> = the area<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula. Substitute.<\/td>\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223846\/CNX_BMath_Figure_09_04_068_img_MW-02.png\" alt=\"The formula A = L times W. The formula is then written again with 32 substituted in for L and 20 substituted in for W\" width=\"310\" height=\"64\"><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]A=640[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong><\/td>\n<td>\n<p>[latex]A\\stackrel{?}{=}640[\/latex]<\/p>\n<p>[latex]32\\cdot 20\\stackrel{?}{=}640[\/latex]<\/p>\n<p>[latex]640=640\\checkmark[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The area of the rectangular playground<br>\nis [latex]640[\/latex] square meters.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question hide_question_numbers=1]288443[\/ohm_question]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>The perimeter of a triangular garden is [latex]24[\/latex] feet. The lengths of two sides are [latex]4[\/latex] feet and [latex]9[\/latex] feet. How long is the third side?<br>\n[reveal-answer q=\"371512\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"371512\"]<\/p>\n<table id=\"eip-id1168466081900\" 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.<\/td>\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223923\/CNX_BMath_Figure_09_04_074_img-01.png\" alt=\"An acute triangle with one side labeled 4 feet, the second side labeled 9 feet, and the third side labeled c. Beneath the triangle, it says P = 24 feet.\" width=\"317\" height=\"188\"><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>length of the third side of a triangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>c<\/em> = the third side<\/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 in the given information.<\/p>\n<\/td>\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223925\/CNX_BMath_Figure_09_04_074_img-02.png\" alt=\"The equation P = a + b + c. The equation is written again with 24 substituted in for P, 4 substituted in for a, and 9 substituted in for b.\" width=\"317\" height=\"67\"><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>\n<p>[latex]24=13+c[\/latex]<\/p>\n<p>[latex]11=c[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 6. <strong>Check.<\/strong><\/p>\n<\/td>\n<td>\n<p>[latex]P=a+b+c[\/latex]<\/p>\n<p>[latex]24\\stackrel{?}{=}4+9+11[\/latex]<\/p>\n<p>[latex]24=24\\checkmark[\/latex]<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The third side is [latex]11[\/latex] feet long.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>A circular sandbox has a radius of [latex]2.5[\/latex] feet. Find the<\/p>\n<ol style=\"list-style-type: decimal;\">\n\t<li>Circumference of the sandbox<\/li>\n\t<li>Area of the sandbox<\/li>\n<\/ol>\n<p>[reveal-answer q=\"247910\"]Show Solution[\/reveal-answer] [hidden-answer a=\"247910\"]<\/p>\n<table>\n<tbody>\n<tr>\n<th>1. Circumference of the sandbox<\/th>\n<\/tr>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224028\/CNX_BMath_Figure_09_05_029_img-01.png\" alt=\"A circle with radius labeled as 2.5 feet\" width=\"159\" height=\"159\"><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the circumference of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>c<\/em> = circumference of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula Substitute<\/td>\n<td>[latex]C=2\\pi r[\/latex] [latex]C=2\\pi \\left(2.5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]C\\approx 2\\left(3.14\\right)\\left(2.5\\right)[\/latex] [latex]C\\approx 15\\text{ft}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong> Does this answer make sense?<\/td>\n<td>Yes. If we draw a square around the circle, its sides would be [latex]5[\/latex] ft (twice the radius), so its perimeter would be [latex]20[\/latex] ft. This is slightly more than the circle's circumference, [latex]15.7[\/latex] ft. <img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224029\/CNX_BMath_Figure_09_05_029_img-02.png\" alt=\"A circle in a red square. The circle's radius is shown as 2.5 feet and the sides of the square are each labeled as 5 feet.\" width=\"206\" height=\"188\"><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The circumference of the sandbox is [latex]15.7[\/latex] feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<th>2. Area of the sandbox<\/th>\n<\/tr>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224028\/CNX_BMath_Figure_09_05_029_img-01.png\" alt=\"A circle with radius labeled as 2.5 feet\" width=\"159\" height=\"159\"><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the area of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>A<\/em> = the area of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula Substitute<\/td>\n<td>[latex]A=\\pi {r}^{2}[\/latex] [latex]A=\\pi{\\left(2.5\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]A\\approx \\left(3.14\\right){\\left(2.5\\right)}^{2}[\/latex] [latex]A\\approx 19.625\\text{ sq. ft}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong> Does this answer make sense?<\/td>\n<td>Yes. If we draw a square around the circle, its sides would be [latex]5[\/latex] ft, as shown in part 1. So the area of the square would be [latex]25[\/latex] sq. ft. This is slightly more than the circle's area, [latex]19.625[\/latex] sq. ft.<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The area of the circle is [latex]19.625[\/latex] square feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>A small globe has a radius of [latex]6[\/latex] centimeters. Find the volume of the globe.&nbsp;<\/p>\n<p><img class=\"wp-image-2307 size-thumbnail aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194506\/pexels-lilartsy-1925535-scaled-1.jpg\" alt=\"Image of globe\" width=\"150\" height=\"150\"><\/p>\n<p>[reveal-answer q=\"133743\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"133743\"]<\/p>\n<p>We will use the formula for calculating the volume of a sphere. In this case, [latex]r=6[\/latex].<\/p>\n<p>So, we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}V=\\frac{4}{3}\\pi r^3\\\\V=\\frac{4}{3}\\pi (6)^3\\\\V=\\frac{4}{3}\\pi (216)\\\\V=\\frac{864}{3}\\pi \\\\V=288\\pi\\\\\\\\V=904.32 \\text{ centimeters}^3\\end{array}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question hide_question_numbers=1]221901[\/ohm_question]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>A rectangular fish tank has a length of [latex]14[\/latex] inches, a height of [latex]17[\/latex] inches, and a width of [latex]9[\/latex] inches. Find its volume.<\/p>\n<p>[reveal-answer q=\"133744\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"133744\"]<\/p>\n<p>We will use the formula for calculating the volume of a rectangular solid. In this case, [latex]l=14[\/latex], [latex]h=17[\/latex], and [latex]w=9[\/latex].<\/p>\n<p>So, we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}V=lwh\\\\V=(14)(9)(17)\\\\V=2142 \\text{ inches}^3\\end{array}[\/latex]<\/p>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question hide_question_numbers=1]221902[\/ohm_question]<\/p>\n<\/section>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use geometric formulas to find the volume, area, and perimeter of shapes in real-life problems<\/li>\n<\/ul>\n<\/section>\n<p>Understanding how to apply geometric formulas is essential for solving practical problems you will encounter in calculus and everyday life. These skills are particularly useful in various physical applications such as determining the mass of objects, calculating work done by variable forces, and finding the hydrostatic force against submerged plates.<\/p>\n<h2>Appling Geometric Formulas to Solve for Volume, Area, and Perimeter<\/h2>\n<p>To effectively apply these geometric formulas, it&#8217;s essential to understand the components of each formula and how they relate to the shapes involved. By mastering these basic principles, you will be better equipped to solve a variety of practical problems in both academic and real-world contexts.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>essential geometric formulas<\/h3>\n<p>To solve practical problems involving geometry, remember the key formulas:<\/p>\n<ul>\n<li><strong>Volume<\/strong>:\n<ul>\n<li>Rectangular Prism: [latex]V = l \\times w \\times h[\/latex]&nbsp;<\/li>\n<li>Cylinder:&nbsp; [latex]V = \\pi r^2 h[\/latex]&nbsp;<\/li>\n<li>Sphere: V = [latex]\\frac{4}{3} \\pi r^3[\/latex]&nbsp;<\/li>\n<\/ul>\n<\/li>\n<li><strong>Area<\/strong>:\n<ul>\n<li>Rectangle: [latex]A = l \\times w[\/latex]<\/li>\n<li>Triangle: [latex]A = \\frac{1}{2} b \\times h[\/latex]&nbsp;<\/li>\n<li>Circle: [latex]A = \\pi r^2[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Perimeter<\/strong>:\n<ul>\n<li>Rectangle: [latex]P = 2l + 2w[\/latex]<\/li>\n<li>Triangle: [latex]P = a + b + c[\/latex]<\/li>\n<li>Circle (Circumference): [latex]C = 2\\pi r[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>Here, [latex]l[\/latex] stands for length, [latex]w[\/latex] for width, [latex]h[\/latex] for height, [latex]r[\/latex] for radius, [latex]b[\/latex] for base (in area of a triangle formula), [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] for the sides of a triangle (in perimeter of a triangle formula), and [latex]\\pi[\/latex] is the constant Pi (approximately [latex]3.14159[\/latex]).<\/p>\n<p>&nbsp;<\/p>\n<\/section>\n<section class=\"textbox questionHelp\"><strong>How to: Solve Volume, Area, and Perimeter Problems<\/strong><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li><strong>Identify the shape:<\/strong> Determine whether you are working with a rectangle, triangle, circle, cylinder, etc.<\/li>\n<li><strong>Choose the appropriate formula:<\/strong> Select the formula that corresponds to the shape and the measurement you need to find (volume, area, or perimeter).<\/li>\n<li><strong>Substitute the given values:<\/strong> Plug in the values provided in the problem into the formula.<\/li>\n<li><strong>Solve the equation:<\/strong> Perform the calculations to find the answer.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox proTip\">\n<p>When working with geometry formulas, we recommend using the following problem-solving strategy when solving.<\/p>\n<p><strong>Problem-Solving Strategy for Geometry Applications<\/strong><\/p>\n<ol id=\"eip-id1170325410595\" class=\"stepwise\">\n<li><strong>Read<\/strong> the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.<\/li>\n<li><strong>Identify<\/strong> what you are looking for.<\/li>\n<li><strong>Name<\/strong> what you are looking for and choose a variable to represent it.<\/li>\n<li><strong>Translate<\/strong> into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.<\/li>\n<li><strong>Solve<\/strong> the equation using good algebra techniques.<\/li>\n<li><strong>Check<\/strong> the answer in the problem and make sure it makes sense.<\/li>\n<li><strong>Answer<\/strong> the question with a complete sentence.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">\n<p>The length of a rectangular playground is [latex]32[\/latex] meters and the width is [latex]20[\/latex] meters. Find the<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Perimeter of the rectangular playground<\/li>\n<li>Area of the rectangular playground<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q172561\">Show Solution<\/button> <\/p>\n<div id=\"q172561\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>\n<table>\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/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\/24223840\/CNX_BMath_Figure_09_04_067_img_MW-01.png\" alt=\"A rectangle with the top and bottom labeled 32 m and the sides labeled 20 m\" width=\"303\" height=\"174\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the perimeter of a rectangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]P[\/latex] = the perimeter<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula. Substitute.<\/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\/24223841\/CNX_BMath_Figure_09_04_067_img_MW-02.png\" alt=\"The formula P = 2L + 2W. The formula is then written again with 32 substituted in for L and 20 substituted in for W\" width=\"524\" height=\"100\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]P=64+40[\/latex] [latex]P=104[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong><\/td>\n<td>\n[latex]p\\stackrel{?}{=}104[\/latex]<br \/>\n[latex]20+32+20+32\\stackrel{?}{=}104[\/latex]<br \/>\n[latex]104=104\\checkmark[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The perimeter of the rectangle is [latex]104[\/latex] meters.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>\n<table>\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/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\/24223845\/CNX_BMath_Figure_09_04_068_img_MW-01.png\" alt=\"A rectangle with the top and bottom labeled 32 m and the sides labeled 20 m\" width=\"310\" height=\"176\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the area of a rectangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>A<\/em> = the area<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula. Substitute.<\/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\/24223846\/CNX_BMath_Figure_09_04_068_img_MW-02.png\" alt=\"The formula A = L times W. The formula is then written again with 32 substituted in for L and 20 substituted in for W\" width=\"310\" height=\"64\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]A=640[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong><\/td>\n<td>\n[latex]A\\stackrel{?}{=}640[\/latex]<br \/>\n[latex]32\\cdot 20\\stackrel{?}{=}640[\/latex]<br \/>\n[latex]640=640\\checkmark[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The area of the rectangular playground<br \/>\nis [latex]640[\/latex] square meters.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288443\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288443&theme=lumen&iframe_resize_id=ohm288443&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox example\">\n<p>The perimeter of a triangular garden is [latex]24[\/latex] feet. The lengths of two sides are [latex]4[\/latex] feet and [latex]9[\/latex] feet. How long is the third side?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q371512\">Show Solution<\/button><\/p>\n<div id=\"q371512\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168466081900\" 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.<\/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\/24223923\/CNX_BMath_Figure_09_04_074_img-01.png\" alt=\"An acute triangle with one side labeled 4 feet, the second side labeled 9 feet, and the third side labeled c. Beneath the triangle, it says P = 24 feet.\" width=\"317\" height=\"188\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>length of the third side of a triangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>c<\/em> = the third side<\/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 in 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\/24223925\/CNX_BMath_Figure_09_04_074_img-02.png\" alt=\"The equation P = a + b + c. The equation is written again with 24 substituted in for P, 4 substituted in for a, and 9 substituted in for b.\" width=\"317\" height=\"67\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>\n[latex]24=13+c[\/latex]<br \/>\n[latex]11=c[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 6. <strong>Check.<\/strong><\/p>\n<\/td>\n<td>\n[latex]P=a+b+c[\/latex]<br \/>\n[latex]24\\stackrel{?}{=}4+9+11[\/latex]<br \/>\n[latex]24=24\\checkmark[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The third side is [latex]11[\/latex] feet long.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>A circular sandbox has a radius of [latex]2.5[\/latex] feet. Find the<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Circumference of the sandbox<\/li>\n<li>Area of the sandbox<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q247910\">Show Solution<\/button> <\/p>\n<div id=\"q247910\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<th>1. Circumference of the sandbox<\/th>\n<\/tr>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/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\/24224028\/CNX_BMath_Figure_09_05_029_img-01.png\" alt=\"A circle with radius labeled as 2.5 feet\" width=\"159\" height=\"159\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the circumference of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>c<\/em> = circumference of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula Substitute<\/td>\n<td>[latex]C=2\\pi r[\/latex] [latex]C=2\\pi \\left(2.5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]C\\approx 2\\left(3.14\\right)\\left(2.5\\right)[\/latex] [latex]C\\approx 15\\text{ft}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong> Does this answer make sense?<\/td>\n<td>Yes. If we draw a square around the circle, its sides would be [latex]5[\/latex] ft (twice the radius), so its perimeter would be [latex]20[\/latex] ft. This is slightly more than the circle&#8217;s circumference, [latex]15.7[\/latex] ft. <img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224029\/CNX_BMath_Figure_09_05_029_img-02.png\" alt=\"A circle in a red square. The circle's radius is shown as 2.5 feet and the sides of the square are each labeled as 5 feet.\" width=\"206\" height=\"188\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The circumference of the sandbox is [latex]15.7[\/latex] feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<th>2. Area of the sandbox<\/th>\n<\/tr>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/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\/24224028\/CNX_BMath_Figure_09_05_029_img-01.png\" alt=\"A circle with radius labeled as 2.5 feet\" width=\"159\" height=\"159\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the area of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>A<\/em> = the area of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula Substitute<\/td>\n<td>[latex]A=\\pi {r}^{2}[\/latex] [latex]A=\\pi{\\left(2.5\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]A\\approx \\left(3.14\\right){\\left(2.5\\right)}^{2}[\/latex] [latex]A\\approx 19.625\\text{ sq. ft}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong> Does this answer make sense?<\/td>\n<td>Yes. If we draw a square around the circle, its sides would be [latex]5[\/latex] ft, as shown in part 1. So the area of the square would be [latex]25[\/latex] sq. ft. This is slightly more than the circle&#8217;s area, [latex]19.625[\/latex] sq. ft.<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The area of the circle is [latex]19.625[\/latex] square feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>A small globe has a radius of [latex]6[\/latex] centimeters. Find the volume of the globe.&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2307 size-thumbnail aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/53\/2025\/02\/13194506\/pexels-lilartsy-1925535-scaled-1.jpg\" alt=\"Image of globe\" width=\"150\" height=\"150\" \/><\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q133743\">Show Solution<\/button><\/p>\n<div id=\"q133743\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will use the formula for calculating the volume of a sphere. In this case, [latex]r=6[\/latex].<\/p>\n<p>So, we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}V=\\frac{4}{3}\\pi r^3\\\\V=\\frac{4}{3}\\pi (6)^3\\\\V=\\frac{4}{3}\\pi (216)\\\\V=\\frac{864}{3}\\pi \\\\V=288\\pi\\\\\\\\V=904.32 \\text{ centimeters}^3\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm221901\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=221901&theme=lumen&iframe_resize_id=ohm221901&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox example\">\n<p>A rectangular fish tank has a length of [latex]14[\/latex] inches, a height of [latex]17[\/latex] inches, and a width of [latex]9[\/latex] inches. Find its volume.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q133744\">Show Solution<\/button><\/p>\n<div id=\"q133744\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will use the formula for calculating the volume of a rectangular solid. In this case, [latex]l=14[\/latex], [latex]h=17[\/latex], and [latex]w=9[\/latex].<\/p>\n<p>So, we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}V=lwh\\\\V=(14)(9)(17)\\\\V=2142 \\text{ inches}^3\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm221902\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=221902&theme=lumen&iframe_resize_id=ohm221902&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":6,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":450,"module-header":"","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\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/454"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/454\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/450"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/454\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=454"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=454"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=454"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=454"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}