{"id":2597,"date":"2025-08-13T18:10:18","date_gmt":"2025-08-13T18:10:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=2597"},"modified":"2026-01-14T16:27:57","modified_gmt":"2026-01-14T16:27:57","slug":"rational-functions-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/rational-functions-background-youll-need-1\/","title":{"raw":"Rational Functions: Background You'll Need 1","rendered":"Rational Functions: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Figure out which values make a rational expression impossible to calculate (like dividing by zero)<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"font-600 text-xl font-bold\">Rational Expressions and Undefined Values<\/h2>\r\n<p class=\"whitespace-pre-wrap break-words\">A rational expression is an algebraic expression that can be written as the quotient of two polynomials, [latex]P(x)[\/latex] and [latex]Q(x)[\/latex], where [latex]Q(x) \\neq 0[\/latex]. It takes the general form:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{P(x)}{Q(x)}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">where [latex]P(x)[\/latex] is the numerator and [latex]Q(x)[\/latex] is the denominator.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">The domain of a rational expression includes all real numbers except those that make the denominator equal to zero. When the denominator equals zero, the expression is undefined. This concept is rooted in the fundamental principle that division by zero is impossible in mathematics.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>rational expressions and undefined values<\/h3>\r\nA rational expression is undefined when its denominator equals zero.\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to: Find the undefined values of a rational expression<\/strong>\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Isolate the denominator.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set the denominator equal to zero.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the resulting equation.<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">The solutions to this equation are the values that make the rational expression undefined.<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Determine the value(s) of [latex]x[\/latex] for which the following rational expression is undefined:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{x^2 - 4}{x - 2}[\/latex]<\/p>\r\n[reveal-answer q=\"138302\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"138302\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the denominator: [latex](x - 2)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set the denominator to zero: [latex]x - 2 = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]:\r\n[latex]x = 2[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the expression is undefined when [latex]x = 2[\/latex].<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Note: At [latex]x = 2[\/latex], both the numerator and denominator equal zero, creating an [pb_glossary id=\"4395\"]indeterminate form[\/pb_glossary] [latex]\\frac{0}{0}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Find the values of [latex]x[\/latex] that make the following rational expression undefined:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{x^2 + 3x}{x^2 - 4}[\/latex]<\/p>\r\n[reveal-answer q=\"298475\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"298475\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the denominator: [latex](x^2 - 4)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set the denominator to zero: [latex]x^2 - 4 = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor the equation: [latex](x + 2)(x - 2) = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]:\r\n[latex]x + 2 = 0[\/latex] or [latex]x - 2 = 0[\/latex]\r\n[latex]x = -2[\/latex] or [latex]x = 2[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the expression is undefined when [latex]x = 2[\/latex] or [latex]x = -2[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Determine all values of [latex]x[\/latex] for which the following rational expression is undefined:<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{2x^2 - 5}{x^3 - x}[\/latex]<\/p>\r\n[reveal-answer q=\"156506\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"156506\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the denominator: [latex]x^3 - x[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Factor the denominator: [latex]x(x^2 - 1) = x(x + 1)(x - 1)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Set each factor to zero and solve:\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">\u00a0[latex]x = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x + 1 = 0[\/latex], so [latex]x = -1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]x - 1 = 0[\/latex], so [latex]x = 1[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the expression is undefined when [latex]x = 0[\/latex], [latex]x = 1[\/latex], and [latex]x = -1[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]318935[\/ohm_question]\r\n\r\n<\/section>Undefined values in rational expressions correspond to vertical asymptotes in their graphs. As [latex]x[\/latex] approaches an undefined value, the expression approaches infinity or negative infinity, creating a vertical line that the graph approaches but never crosses.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">For example, in the graph of [latex]y = \\frac{1}{x - 2}[\/latex]:<\/p>\r\n\r\n\r\n[caption id=\"attachment_4394\" align=\"aligncenter\" width=\"867\"]<img class=\"wp-image-4394 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/02155823\/Screenshot-2024-10-02-115805.png\" alt=\"Graph of y = \\frac{1}{x - 2}\" width=\"867\" height=\"833\" \/> Graph of y = 1 \/ (x-2)[\/caption]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches [latex]2[\/latex] from the left, [latex]y[\/latex] approaches positive infinity.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches [latex]2[\/latex] from the right, [latex]y[\/latex] approaches negative infinity.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The line [latex]x = 2[\/latex] is a vertical asymptote of the graph.<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">\r\n<p class=\"font-600 text-xl font-bold\"><strong>Common Mistakes to Avoid<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Forgetting to check for undefined values before simplifying.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Assuming only linear terms in the denominator can cause undefined values.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Neglecting to factor completely when dealing with higher degree polynomials.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Confusing zero values of the numerator with undefined values.<\/li>\r\n<\/ol>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\">Figure out which values make a rational expression impossible to calculate (like dividing by zero)<\/span><\/li>\n<\/ul>\n<\/section>\n<h2 class=\"font-600 text-xl font-bold\">Rational Expressions and Undefined Values<\/h2>\n<p class=\"whitespace-pre-wrap break-words\">A rational expression is an algebraic expression that can be written as the quotient of two polynomials, [latex]P(x)[\/latex] and [latex]Q(x)[\/latex], where [latex]Q(x) \\neq 0[\/latex]. It takes the general form:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{P(x)}{Q(x)}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">where [latex]P(x)[\/latex] is the numerator and [latex]Q(x)[\/latex] is the denominator.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">The domain of a rational expression includes all real numbers except those that make the denominator equal to zero. When the denominator equals zero, the expression is undefined. This concept is rooted in the fundamental principle that division by zero is impossible in mathematics.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>rational expressions and undefined values<\/h3>\n<p>A rational expression is undefined when its denominator equals zero.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How to: Find the undefined values of a rational expression<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Isolate the denominator.<\/li>\n<li class=\"whitespace-normal break-words\">Set the denominator equal to zero.<\/li>\n<li class=\"whitespace-normal break-words\">Solve the resulting equation.<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">The solutions to this equation are the values that make the rational expression undefined.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Determine the value(s) of [latex]x[\/latex] for which the following rational expression is undefined:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{x^2 - 4}{x - 2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q138302\">Show Answer<\/button><\/p>\n<div id=\"q138302\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the denominator: [latex](x - 2)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Set the denominator to zero: [latex]x - 2 = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]:<br \/>\n[latex]x = 2[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the expression is undefined when [latex]x = 2[\/latex].<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Note: At [latex]x = 2[\/latex], both the numerator and denominator equal zero, creating an <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_2597_4395\">indeterminate form<\/a> [latex]\\frac{0}{0}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Find the values of [latex]x[\/latex] that make the following rational expression undefined:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{x^2 + 3x}{x^2 - 4}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q298475\">Show Answer<\/button><\/p>\n<div id=\"q298475\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the denominator: [latex](x^2 - 4)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Set the denominator to zero: [latex]x^2 - 4 = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Factor the equation: [latex](x + 2)(x - 2) = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve for [latex]x[\/latex]:<br \/>\n[latex]x + 2 = 0[\/latex] or [latex]x - 2 = 0[\/latex]<br \/>\n[latex]x = -2[\/latex] or [latex]x = 2[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the expression is undefined when [latex]x = 2[\/latex] or [latex]x = -2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Determine all values of [latex]x[\/latex] for which the following rational expression is undefined:<\/p>\n<p class=\"whitespace-pre-wrap break-words\" style=\"text-align: center;\">[latex]\\frac{2x^2 - 5}{x^3 - x}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q156506\">Show Answer<\/button><\/p>\n<div id=\"q156506\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the denominator: [latex]x^3 - x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Factor the denominator: [latex]x(x^2 - 1) = x(x + 1)(x - 1)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Set each factor to zero and solve:\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">\u00a0[latex]x = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x + 1 = 0[\/latex], so [latex]x = -1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]x - 1 = 0[\/latex], so [latex]x = 1[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the expression is undefined when [latex]x = 0[\/latex], [latex]x = 1[\/latex], and [latex]x = -1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm318935\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=318935&theme=lumen&iframe_resize_id=ohm318935&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/section>\n<p>Undefined values in rational expressions correspond to vertical asymptotes in their graphs. As [latex]x[\/latex] approaches an undefined value, the expression approaches infinity or negative infinity, creating a vertical line that the graph approaches but never crosses.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">For example, in the graph of [latex]y = \\frac{1}{x - 2}[\/latex]:<\/p>\n<figure id=\"attachment_4394\" aria-describedby=\"caption-attachment-4394\" style=\"width: 867px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4394 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/06\/02155823\/Screenshot-2024-10-02-115805.png\" alt=\"Graph of y = \\frac{1}{x - 2}\" width=\"867\" height=\"833\" \/><figcaption id=\"caption-attachment-4394\" class=\"wp-caption-text\">Graph of y = 1 \/ (x-2)<\/figcaption><\/figure>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches [latex]2[\/latex] from the left, [latex]y[\/latex] approaches positive infinity.<\/li>\n<li class=\"whitespace-normal break-words\">As [latex]x[\/latex] approaches [latex]2[\/latex] from the right, [latex]y[\/latex] approaches negative infinity.<\/li>\n<li class=\"whitespace-normal break-words\">The line [latex]x = 2[\/latex] is a vertical asymptote of the graph.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">\n<p class=\"font-600 text-xl font-bold\"><strong>Common Mistakes to Avoid<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Forgetting to check for undefined values before simplifying.<\/li>\n<li class=\"whitespace-normal break-words\">Assuming only linear terms in the denominator can cause undefined values.<\/li>\n<li class=\"whitespace-normal break-words\">Neglecting to factor completely when dealing with higher degree polynomials.<\/li>\n<li class=\"whitespace-normal break-words\">Confusing zero values of the numerator with undefined values.<\/li>\n<\/ol>\n<\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_2597_4395\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_2597_4395\"><div tabindex=\"-1\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the length of a circular arc.<\/li>\n<li>Find the area of a sector of a circle.<\/li>\n<li>Use linear and angular speed to describe motion on a circular path.<\/li>\n<\/ul>\n<\/section>\n<h2>Finding the Length of a Circular Arc<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>The length of a circular arc tells us how far along the circle\u2019s edge we travel when sweeping out an angle. The key is that the angle must be measured in radians, because radians directly connect an angle to the arc length. If [latex]\\theta[\/latex] is the central angle in radians and [latex]r[\/latex] is the circle\u2019s radius, then the arc length is given by [latex]s = r\\theta[\/latex]. This formula works because radians are defined as arc length divided by radius<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">.<\/strong><\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Finding Arc Length<\/strong><\/p>\n<ol>\n<li>Use the formula: [latex]s = r\\theta[\/latex], where [latex]s[\/latex] is arc length, [latex]r[\/latex] is radius, and [latex]\\theta[\/latex] is in radians.<\/li>\n<li>Covert first if needed: If the angle is given in degrees, convert to radians before using the formula.<\/li>\n<li>Check the units: The arc length will be in the same unit as the radius (e.g., if [latex]r[\/latex] is in cm, [latex]s[\/latex] will be in cm).<\/li>\n<li>Fraction of the circle: You can also think of arc length as a fraction of the whole circumference:\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]s = \\dfrac{\\theta}{2\\pi}\\cdot (2\\pi r)[\/latex].<\/li>\n<\/ol>\n<\/li>\n<li>Real-world connection: Arc length is like the \u201cdistance walked\u201d along the edge of the circle \u2014 useful for wheels, gears, and circular tracks.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A city bus navigates a roundabout of radius [latex]14[\/latex]m, turning through [latex]95^\\circ[\/latex]. Find the arc length [latex]s[\/latex].[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q725997\">Hint<\/button><\/p>\n<div id=\"q725997\" class=\"hidden-answer\" style=\"display: none\">[latex]\\\\\\\\[\/latex]Convert [latex]95^\\circ[\/latex] to radians.[latex]\\\\\\\\[\/latex][latex]\\\\\\\\[\/latex]<br \/>\n[latex]s = r\\theta[\/latex] with [latex]\\theta[\/latex] in radians[latex]\\\\\\\\[\/latex]\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q702833\">Show Answer<\/button><\/p>\n<div id=\"q702833\" class=\"hidden-answer\" style=\"display: none\">[latex]95^\\circ \\Rightarrow[\/latex] radians:<br \/>\n[latex]\\begin{aligned}  95^\\circ\\cdot \\dfrac{\\pi}{180} &= \\dfrac{95\\pi}{180} \\\\  &= \\dfrac{19\\pi}{36} \\text { radians}  \\end{aligned}[\/latex][latex]\\\\\\\\[\/latex]<br \/>\n[latex]\\begin{aligned}s &= r\\theta \\\\ &= 14 \\cdot \\dfrac{19\\pi}{36} \\\\&= \\dfrac{133\\pi}{18} \\\\ &\\approx 23.21 \\text { m}\\end{aligned}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A robot vacuum hugs a circular coffee table, following an arc with radius [latex]0.70[\/latex] m and central angle [latex]1.9[\/latex] radians. Find [latex]s[\/latex].[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q268488\">Hint<\/button><\/p>\n<div id=\"q268488\" class=\"hidden-answer\" style=\"display: none\">Angle is already in radians, and the radius is given.[latex]\\\\\\\\[\/latex]Remember: [latex]s = r\\theta[\/latex][latex]\\\\\\\\[\/latex]<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q939558\">Show Answer<\/button><\/p>\n<div id=\"q939558\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{aligned}  s &= 0.70 \\cdot 1.9 \\\\ &= 1.33 \\end{aligned}[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">In the planetarium, a laser dot sweeps along the dome at radius [latex]11[\/latex]m through angle [latex]\\dfrac{7\\pi}{20}[\/latex]. Find [latex]s[\/latex] in exact form.[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q828216\">Hint<\/button><\/p>\n<div id=\"q828216\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{aligned} s &= r\\theta \\\\ &= 11 \\cdot \\dfrac{7\\pi}{20}\\end{aligned}[\/latex][latex]\\\\\\\\[\/latex]Keep [latex]\\pi[\/latex] for an exact answer.[latex]\\\\\\\\[\/latex]<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q97179\">Show Answer<\/button><\/p>\n<div id=\"q97179\" class=\"hidden-answer\" style=\"display: none\">[latex]s = \\dfrac{77\\pi}{20}[\/latex]<\/div>\n<\/div>\n<p>\u00a0<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox watchIt\">Insert video here<\/section>\n<\/div>\n<h2>Finding the Area of a Sector<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>A sector of a circle is like a <strong data-start=\"195\" data-end=\"213\">slice of pizza<\/strong> \u2014 it\u2019s the region between two radii and the arc that connects them. The area of that slice depends on how big the angle is and how large the circle is. If the radius is [latex]r[\/latex] and the central angle is [latex]\\theta[\/latex] in radians, then the area of the sector is given by [latex]A = \\tfrac{1}{2}r^2\\theta[\/latex]. This formula works because radians directly connect angle size to the fraction of the circle\u2019s area.<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Finding Area of a Sector<\/strong><\/p>\n<ol>\n<li>Use the formula: [latex]A = \\tfrac{1}{2}r^2\\theta[\/latex], where [latex]\\theta[\/latex] is in radians.<\/li>\n<li>Convert if necessary: If the angle is in degrees, convert to radians before plugging it in.<\/li>\n<li>Check the units: The area will be in square units (e.g., [latex]\\text{cm}^2[\/latex]) if the radius is in cm.<\/li>\n<li>Fraction of the whole circle: The formula also comes from [latex]\\dfrac{\\theta}{2\\pi}\\cdot \\pi r^2[\/latex]; the fraction of the circle\u2019s total area.<\/li>\n<li>Think pizza or pie: A small angle gives a skinny slice, a big angle gives a bigger slice \u2014 the formula measures the \u201csize of the slice.\u201d<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A sprinkler waters a sector of radius [latex]6[\/latex]m across [latex]110^\\circ[\/latex]. Find the watered area.[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q710677\">Hint<\/button><\/p>\n<div id=\"q710677\" class=\"hidden-answer\" style=\"display: none\">You know the radius. Convert [latex]110^\\circ[\/latex] to radians, and then plug into the formula.[latex]\\\\\\\\[\/latex]<br \/>\n[latex]A=\\dfrac{1}{2}r^2\\theta[\/latex]<\/div>\n<\/div>\n<p>[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q256399\">Show Answer<\/button><\/p>\n<div id=\"q256399\" class=\"hidden-answer\" style=\"display: none\">[latex]\\begin{aligned} A &= \\dfrac{1}{2}(6)^2 \\cdot \\dfrac{11\\pi}{18} \\\\ &= 18 \\cdot \\dfrac{11\\pi}{18} \\\\ &= 11\\pi \\\\ &\\approx 34.56 \\text { m^2}\\end{aligned}[\/latex] <\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">At a night market, a spotlight covers a wedge of radius [latex]12[\/latex] m with angle [latex]0.90[\/latex] radians. Find the illuminated area.[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q446876\">Hint<\/button><\/p>\n<div id=\"q446876\" class=\"hidden-answer\" style=\"display: none\">You know the radius and the radians.[latex]\\\\\\\\[\/latex]Remember: [latex]A = \\dfrac{1}{2} r^2\\theta[\/latex]<\/div>\n<\/div>\n<p>[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q680566\">Show Answer<\/button><\/p>\n<div id=\"q680566\" class=\"hidden-answer\" style=\"display: none\">[latex]A = \\dfrac{1}{2} \\cdot (12)^2 \\cdot 0.90[\/latex][latex]\\\\\\\\[\/latex][latex]= \\dfrac{1}{2} \\cdot 144 \\cdot 0.90[\/latex][latex]\\\\\\\\[\/latex][latex]= 64.8[\/latex] m[latex]^2[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A beadwork medallion shows a sector of radius [latex]8[\/latex] cm with angle [latex]\\dfrac{5\\pi}{6}[\/latex]. Find the sector area in exact form.[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q175607\">Hint<\/button><\/p>\n<div id=\"q175607\" class=\"hidden-answer\" style=\"display: none\">[latex]A = \\dfrac{1}{2} r^2\\theta[\/latex][latex]\\\\\\\\[\/latex][latex]= \\dfrac{1}{2} \\cdot 64 \\cdot \\dfrac{5\\pi}{6}[\/latex][latex]\\\\\\\\[\/latex]Do not change [latex]\\pi[\/latex] to a decimal.<\/div>\n<\/div>\n<p>[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q209703\">Show Answer<\/button><\/p>\n<div id=\"q209703\" class=\"hidden-answer\" style=\"display: none\">[latex]A = 32 \\cdot \\dfrac{5\\pi}{6}[\/latex][latex]\\\\\\\\[\/latex][latex]= \\dfrac{80\\pi}{3}[\/latex] cm[latex]^2[\/latex][latex]\\\\\\\\[\/latex][latex]\\approx 83.78[\/latex] cm[latex]^2[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox watchIt\">Insert video here<\/section>\n<\/div>\n<h2>Describing Motion on a Circular Path<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p>When an object moves along a circular path, we can describe its motion in two connected ways: <strong data-start=\"281\" data-end=\"298\">angular speed<\/strong> and <strong data-start=\"303\" data-end=\"319\">linear speed<\/strong>. Angular speed measures <strong data-start=\"344\" data-end=\"374\">how fast the angle changes<\/strong> (in radians per unit of time), while linear speed measures <strong data-start=\"434\" data-end=\"481\">how fast the distance along the arc changes<\/strong> (in distance per unit of time). The two are tied together by the radius:<br data-start=\"554\" data-end=\"557\" \/>[latex]v = r\\omega[\/latex], where [latex]v[\/latex] is linear speed, [latex]r[\/latex] is the radius, and [latex]\\omega[\/latex] is angular speed. This relationship lets us move between \u201cspinning speed\u201d (angular) and \u201ctraveling speed\u201d (linear).<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Finding Area of a Sector<\/strong><\/p>\n<ol>\n<li>Linear speed formula: [latex]\\omega = \\dfrac{\\theta}{t}[\/latex], where [latex]\\theta[\/latex] is in radians and [latex]t[\/latex] is time.<\/li>\n<li>Angular speed formula: [latex]v = \\dfrac{s}{t}[\/latex], where [latex]s[\/latex] is the arc length traveled.<\/li>\n<li>Connect them: Use [latex]s = r\\theta[\/latex] to link the two formulas, giving [latex]v = r\\omega[\/latex].<\/li>\n<li>Units Matter:\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Linear speed: feet per second, meters per second, etc.<\/li>\n<li>Angular speed: radians per second (or per minute).<\/li>\n<\/ol>\n<\/li>\n<li>Think Real-World: On a spinning wheel, all points share the same angular speed, but points farther from the center move faster linearly.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A smartwatch second hand has length [latex]18[\/latex] mm. Find its angular speed [latex]\\omega[\/latex] rad\/s and tip speed [latex]v[\/latex] m\/s.[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q138196\">Hint<\/button><\/p>\n<div id=\"q138196\" class=\"hidden-answer\" style=\"display: none\">[latex]\\omega = \\dfrac{2\\pi}{60}[\/latex] rad\/s[latex]\\\\\\\\[\/latex]<br \/>\n[latex]r = 18[\/latex] mm [latex]= 0.018[\/latex] m[latex]\\\\\\\\[\/latex]<br \/>\n[latex]v = r\\omega[\/latex]<\/div>\n<\/div>\n<p>[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q6300\">Show Answer<\/button><\/p>\n<div id=\"q6300\" class=\"hidden-answer\" style=\"display: none\">[latex]\\omega = \\dfrac{2\\pi}{60} = \\dfrac{\\pi}{30}[\/latex] rad\/s[latex]\\approx 0.1047[\/latex][latex]\\\\\\\\[\/latex]<br \/>\n[latex]v = 0.018 \\cdot \\dfrac{\\pi}{30} = \\dfrac{3\\pi}{5000}[\/latex] m\/s[latex]\\approx 1.885 \\cdot 10^{-3}[\/latex] m\/s\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">A wind turbine (blade length [latex]27[\/latex] m spins at [latex]12[\/latex] RPM. Find [latex]\\omega[\/latex] rad\/s and the tip speed [latex]v[\/latex] m\/s.[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q109771\">Hint<\/button><\/p>\n<div id=\"q109771\" class=\"hidden-answer\" style=\"display: none\">[latex]\\omega = 2\\pi \\cdot \\dfrac{\\text{RPM}}{60} = \\dfrac{2\\pi}{5}[\/latex] rad\/s[latex]\\\\\\\\[\/latex][latex]v=r\\omega.[\/latex]<\/div>\n<\/div>\n<p>[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q367369\">Show Answer<\/button><\/p>\n<div id=\"q367369\" class=\"hidden-answer\" style=\"display: none\">[latex]\\omega = \\dfrac{2\\pi}{5}[\/latex] rad\/s [latex]\\approx 1.257[\/latex][latex]\\\\\\\\[\/latex]<br \/>\n[latex]v=27 \\cdot \\dfrac{2\\pi}{5} = \\dfrac{54\\pi}{5}[\/latex] m\/s [latex]\\approx 33.93[\/latex] m\/s.<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox example\">In an aim-trainer, a circular targe rotates with angular speed [latex]1.8[\/latex] rad\/s. A marker sits [latex]0.75[\/latex] m from the center. Find its linear speed [latex]v[\/latex].[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q416218\">Hint<\/button><\/p>\n<div id=\"q416218\" class=\"hidden-answer\" style=\"display: none\">[latex]v = r\\omega = 0.75 \\cdot 1.8.[\/latex]<\/div>\n<\/div>\n<p>[latex]\\\\\\\\[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q938851\">Show Answer<\/button><\/p>\n<div id=\"q938851\" class=\"hidden-answer\" style=\"display: none\">[latex]v = 1.35[\/latex] m\/s[\/latex]<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p>&nbsp;<\/p>\n<div>\n<section class=\"textbox watchIt\">Insert video here<\/section>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">In the following video you will see how to calculate arc length and area of a sector of a circle.https:\/\/youtu.be\/zD4CsKIYEHo<\/section>\n<section aria-label=\"Watch It\">\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">https:\/\/youtu.be\/bfWkgA5GSE0<\/section>\n<\/section>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":67,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":508,"module-header":"background_you_need","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\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2597"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2597\/revisions"}],"predecessor-version":[{"id":5343,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2597\/revisions\/5343"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/508"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2597\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2597"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2597"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2597"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2597"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}