{"id":3892,"date":"2021-05-20T18:29:55","date_gmt":"2021-05-20T18:29:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-related-rates\/"},"modified":"2021-07-03T18:59:56","modified_gmt":"2021-07-03T18:59:56","slug":"review-for-related-rates","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-related-rates\/","title":{"raw":"Skills Review for Related Rates","rendered":"Skills Review for Related Rates"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate the volume of a cone&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Calculate the volume of a cone<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate the volume of a sphere&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Calculate the volume of a sphere<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate the volume of rectangular solid&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Calculate the volume of rectangular solid<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the Pythagorean theorem to find the unknown side length of a right triangle&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Use the Pythagorean theorem to find the unknown side length of a right triangle<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the distance, rate, and time formula&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,16573901],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Use the distance, rate, and time formula<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Related Rates section, we will take derivatives of various mathematical formulas. Here we will review the formulas for calculating volume of a cone, volume of a sphere, and volume of a rectangular solid. We will also briefly review the Pythagorean Theorem and the distance, rate, and time formula.\r\n<h2>Find the Volume of a Cone<\/h2>\r\nThe <b>volume of a cone<\/b>\u00a0can be found using the following formula:\r\n<p style=\"text-align: center;\">[latex]V=\\frac{1}{3}\\pi r^2h[\/latex]<\/p>\r\nwhere [latex]r[\/latex] is the radius of the circular base of the cone and [latex]h[\/latex] is the height of the cone.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Volume of A Cone<\/h3>\r\nFind the volume of a cone with height 6 inches and a base with radius 2 inches.\r\n\r\n[reveal-answer q=\"133742\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133742\"]\r\n\r\nWe will use the formula for calculating the volume of a cone. In this case, [latex]h=6[\/latex] and [latex]r=2[\/latex].\r\n\r\nSo, we have:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}V=\\frac{1}{3}\\pi r^2h\\\\V=\\frac{1}{3}\\pi (2)^2(6)\\\\V=\\frac{1}{3}\\pi (4)(6)\\\\V=\\frac{1}{3}\\pi (24)\\\\V=\\frac{24}{3}\\pi\\\\V=8\\pi\\\\V=25.12 \\text{ inches}^3\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]146818[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Find the Volume of a Sphere<\/h2>\r\nThe <b>volume of a sphere<\/b>\u00a0can be found using the following formula:\r\n<p style=\"text-align: center;\">[latex]V=\\frac{4}{3}\\pi r^3[\/latex]<\/p>\r\nwhere [latex]r[\/latex] is the radius.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Volume of A Sphere<\/h3>\r\nFind the volume of a sphere with radius 6 centimeters.\r\n\r\n[reveal-answer q=\"133743\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133743\"]\r\n\r\nWe will use the formula for calculating the volume of a sphere. In this case, [latex]r=6[\/latex].\r\n\r\nSo, we have:\r\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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]221901[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Find the Volume of a Rectangular Solid<\/h2>\r\nA rectangular solid has a length, width, and height. The\u00a0<b>volume of a rectangular solid<\/b>\u00a0can be found using the following formula:\r\n<p style=\"text-align: center;\">[latex]V=lwh[\/latex]<\/p>\r\nwhere [latex]l[\/latex] is the length, [latex]w[\/latex] is the width, and [latex]h[\/latex] is the height.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Volume of A Rectangular solid<\/h3>\r\nFind the volume of a rectangular solid with length 14 inches, height 17 inches, and width 9 inches.\r\n\r\n[reveal-answer q=\"133744\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133744\"]\r\n\r\nWe 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].\r\n\r\nSo, we have:\r\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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]221902[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use the Pythagorean Theorem<\/h2>\r\nThe <b>Pythagorean theorem<\/b>\u00a0is a statement about the sides of a right triangle.\u00a0One of the angles of a right triangle is always equal to\u00a0[latex]90[\/latex] degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite the right angle, and it is always the longest side.\r\n\r\n[caption id=\"attachment_4904\" align=\"alignleft\" width=\"205\"]<img class=\"wp-image-4904\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/20161955\/Screen-Shot-2016-06-14-at-8.32.56-PM-268x300.png\" alt=\"right triangle labeled with the longest length = a, and the other two b and c.\" width=\"205\" height=\"229\" \/> A right triangle with sides labeled.[\/caption]\r\n<p style=\"text-align: left;\">The Pythagorean theorem is often used to find unknown lengths of the sides of right triangles. If the longest leg of a right triangle is labeled c, and the other two a, and b as in the image on the left, \u00a0The Pythagorean Theorem states that<\/p>\r\n<p style=\"text-align: center;\">[latex]a^2+b^2=c^2[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Given enough information, we can solve for an unknown length.<\/p>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<p style=\"text-align: left;\">In each of the following examples, we will use the Pythagorean theorem to find a missing side of a right triangle.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using The Pythagorean Theorem to Find The Length of the Hypotenuse<\/h3>\r\nA\u00a0right triangle has legs measuring 6 and 7 centimeters. Find the length of the hypotenuse.\r\n\r\n[reveal-answer q=\"133740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133740\"]\r\n\r\nWe were given the lengths of two legs of a right triangle. In the Pythagorean theorem, it does not matter which leg measurement is used for [latex]a[\/latex] and which leg measurement is used for [latex]b[\/latex]. Our goal is to find the [latex]c[\/latex], the hypotenuse.\r\n\r\nSo, we have:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}a^2+b^2=c^2\\\\\\left(6\\right)^2+\\left(7\\right)^2=c\\\\36+49=c^2\\\\85=c^2\\\\\\sqrt{85}=\\sqrt{c^2}\\\\\\sqrt{85}=c\\\\c=\\sqrt{85}\\\\c=9.22 \\text{ centimeters}\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]221904[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using The Pythagorean Theorem to Find The Length of A Leg<\/h3>\r\nA\u00a0right triangle has a leg measuring 10 inches. The hypotenuse measures 12 inches. Find the length of the other leg.\r\n\r\n[reveal-answer q=\"133741\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133741\"]\r\n\r\nIn the Pythagorean theorem, it does not matter which leg is assigned the length of 10. In this case, we will let [latex]a=10[\/latex] and [latex]b[\/latex] be the unknown leg length. We also know the length of the hypotenuse, so [latex]c=12[\/latex].\r\n\r\nSo, we have:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}a^2+b^2=c^2\\\\\\left(10\\right)^2+b^2=(12)^2\\\\100+b^2=144\\\\b^2=44\\\\\\sqrt{b^2}=\\sqrt{44}\\\\b=\\sqrt{44}\\\\b=6.63 \\text{ inches}\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]221903[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Use the Distance, Rate, and Time Formula<\/h2>\r\nFor an object moving at a uniform (constant) rate, the distance, [latex]d[\/latex], traveled is:\r\n<p style=\"text-align: center;\">[latex]d=rt[\/latex]<\/p>\r\nwhere [latex]r[\/latex] is the rate and [latex]t[\/latex] is the time.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Distance, Rate, and Time Formula<\/h3>\r\nJenny rides her bike at a uniform rate of 10 miles per hour for 3 hours. How far has she traveled?.\r\n\r\n[reveal-answer q=\"133745\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"133745\"]\r\n\r\nIn the distance, rate, and time formula, let [latex]r=10[\/latex] and [latex]t=3[\/latex].\r\n\r\nSo, we have:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}d=rt\\\\d=(10)(3)\\\\d=30 \\text{ miles}\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]221905[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate the volume of a cone&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Calculate the volume of a cone<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate the volume of a sphere&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Calculate the volume of a sphere<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Calculate the volume of rectangular solid&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Calculate the volume of rectangular solid<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the Pythagorean theorem to find the unknown side length of a right triangle&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Use the Pythagorean theorem to find the unknown side length of a right triangle<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the distance, rate, and time formula&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,16573901],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Use the distance, rate, and time formula<\/span><\/li>\n<\/ul>\n<\/div>\n<p>In the Related Rates section, we will take derivatives of various mathematical formulas. Here we will review the formulas for calculating volume of a cone, volume of a sphere, and volume of a rectangular solid. We will also briefly review the Pythagorean Theorem and the distance, rate, and time formula.<\/p>\n<h2>Find the Volume of a Cone<\/h2>\n<p>The <b>volume of a cone<\/b>\u00a0can be found using the following formula:<\/p>\n<p style=\"text-align: center;\">[latex]V=\\frac{1}{3}\\pi r^2h[\/latex]<\/p>\n<p>where [latex]r[\/latex] is the radius of the circular base of the cone and [latex]h[\/latex] is the height of the cone.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Volume of A Cone<\/h3>\n<p>Find the volume of a cone with height 6 inches and a base with radius 2 inches.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133742\">Show Solution<\/span><\/p>\n<div id=\"q133742\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will use the formula for calculating the volume of a cone. In this case, [latex]h=6[\/latex] and [latex]r=2[\/latex].<\/p>\n<p>So, we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}V=\\frac{1}{3}\\pi r^2h\\\\V=\\frac{1}{3}\\pi (2)^2(6)\\\\V=\\frac{1}{3}\\pi (4)(6)\\\\V=\\frac{1}{3}\\pi (24)\\\\V=\\frac{24}{3}\\pi\\\\V=8\\pi\\\\V=25.12 \\text{ inches}^3\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146818\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146818&theme=oea&iframe_resize_id=ohm146818&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Find the Volume of a Sphere<\/h2>\n<p>The <b>volume of a sphere<\/b>\u00a0can be found using the following formula:<\/p>\n<p style=\"text-align: center;\">[latex]V=\\frac{4}{3}\\pi r^3[\/latex]<\/p>\n<p>where [latex]r[\/latex] is the radius.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Volume of A Sphere<\/h3>\n<p>Find the volume of a sphere with radius 6 centimeters.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133743\">Show Solution<\/span><\/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<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm221901\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=221901&theme=oea&iframe_resize_id=ohm221901&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Find the Volume of a Rectangular Solid<\/h2>\n<p>A rectangular solid has a length, width, and height. The\u00a0<b>volume of a rectangular solid<\/b>\u00a0can be found using the following formula:<\/p>\n<p style=\"text-align: center;\">[latex]V=lwh[\/latex]<\/p>\n<p>where [latex]l[\/latex] is the length, [latex]w[\/latex] is the width, and [latex]h[\/latex] is the height.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Volume of A Rectangular solid<\/h3>\n<p>Find the volume of a rectangular solid with length 14 inches, height 17 inches, and width 9 inches.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133744\">Show Solution<\/span><\/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<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm221902\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=221902&theme=oea&iframe_resize_id=ohm221902&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use the Pythagorean Theorem<\/h2>\n<p>The <b>Pythagorean theorem<\/b>\u00a0is a statement about the sides of a right triangle.\u00a0One of the angles of a right triangle is always equal to\u00a0[latex]90[\/latex] degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite the right angle, and it is always the longest side.<\/p>\n<div id=\"attachment_4904\" style=\"width: 215px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4904\" class=\"wp-image-4904\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/20161955\/Screen-Shot-2016-06-14-at-8.32.56-PM-268x300.png\" alt=\"right triangle labeled with the longest length = a, and the other two b and c.\" width=\"205\" height=\"229\" \/><\/p>\n<p id=\"caption-attachment-4904\" class=\"wp-caption-text\">A right triangle with sides labeled.<\/p>\n<\/div>\n<p style=\"text-align: left;\">The Pythagorean theorem is often used to find unknown lengths of the sides of right triangles. If the longest leg of a right triangle is labeled c, and the other two a, and b as in the image on the left, \u00a0The Pythagorean Theorem states that<\/p>\n<p style=\"text-align: center;\">[latex]a^2+b^2=c^2[\/latex]<\/p>\n<p style=\"text-align: left;\">Given enough information, we can solve for an unknown length.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left;\">In each of the following examples, we will use the Pythagorean theorem to find a missing side of a right triangle.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using The Pythagorean Theorem to Find The Length of the Hypotenuse<\/h3>\n<p>A\u00a0right triangle has legs measuring 6 and 7 centimeters. Find the length of the hypotenuse.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133740\">Show Solution<\/span><\/p>\n<div id=\"q133740\" class=\"hidden-answer\" style=\"display: none\">\n<p>We were given the lengths of two legs of a right triangle. In the Pythagorean theorem, it does not matter which leg measurement is used for [latex]a[\/latex] and which leg measurement is used for [latex]b[\/latex]. Our goal is to find the [latex]c[\/latex], the hypotenuse.<\/p>\n<p>So, we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}a^2+b^2=c^2\\\\\\left(6\\right)^2+\\left(7\\right)^2=c\\\\36+49=c^2\\\\85=c^2\\\\\\sqrt{85}=\\sqrt{c^2}\\\\\\sqrt{85}=c\\\\c=\\sqrt{85}\\\\c=9.22 \\text{ centimeters}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm221904\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=221904&theme=oea&iframe_resize_id=ohm221904&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using The Pythagorean Theorem to Find The Length of A Leg<\/h3>\n<p>A\u00a0right triangle has a leg measuring 10 inches. The hypotenuse measures 12 inches. Find the length of the other leg.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133741\">Show Solution<\/span><\/p>\n<div id=\"q133741\" class=\"hidden-answer\" style=\"display: none\">\n<p>In the Pythagorean theorem, it does not matter which leg is assigned the length of 10. In this case, we will let [latex]a=10[\/latex] and [latex]b[\/latex] be the unknown leg length. We also know the length of the hypotenuse, so [latex]c=12[\/latex].<\/p>\n<p>So, we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}a^2+b^2=c^2\\\\\\left(10\\right)^2+b^2=(12)^2\\\\100+b^2=144\\\\b^2=44\\\\\\sqrt{b^2}=\\sqrt{44}\\\\b=\\sqrt{44}\\\\b=6.63 \\text{ inches}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm221903\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=221903&theme=oea&iframe_resize_id=ohm221903&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Use the Distance, Rate, and Time Formula<\/h2>\n<p>For an object moving at a uniform (constant) rate, the distance, [latex]d[\/latex], traveled is:<\/p>\n<p style=\"text-align: center;\">[latex]d=rt[\/latex]<\/p>\n<p>where [latex]r[\/latex] is the rate and [latex]t[\/latex] is the time.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Distance, Rate, and Time Formula<\/h3>\n<p>Jenny rides her bike at a uniform rate of 10 miles per hour for 3 hours. How far has she traveled?.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q133745\">Show Solution<\/span><\/p>\n<div id=\"q133745\" class=\"hidden-answer\" style=\"display: none\">\n<p>In the distance, rate, and time formula, let [latex]r=10[\/latex] and [latex]t=3[\/latex].<\/p>\n<p>So, we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}d=rt\\\\d=(10)(3)\\\\d=30 \\text{ miles}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm221905\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=221905&theme=oea&iframe_resize_id=ohm221905&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3892\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision . <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Modification and Revision \",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra 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