{"id":234250,"date":"2025-06-14T03:02:17","date_gmt":"2025-06-14T03:02:17","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=234250"},"modified":"2025-06-14T03:02:19","modified_gmt":"2025-06-14T03:02:19","slug":"for-questions-13-and-14-find-the-length-of-each-bolded-arc","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/14\/for-questions-13-and-14-find-the-length-of-each-bolded-arc\/","title":{"rendered":"For questions 13 and 14, find the length of each bolded arc."},"content":{"rendered":"\n<p><br>For questions 13 and 14, find the length of each bolded arc. Round to the nearest hundredth. 13.<\/p>\n\n\n\n<p>14.<\/p>\n\n\n\n<ol start=\"13\" class=\"wp-block-list\">\n<li>]]\n<ol start=\"14\" class=\"wp-block-list\">\n<li>75.9 15. Using the circle below, find each arc length. Round to the nearest hundredth. a)<br>b)<br>20 16. A merry-go-round has a radius of feet. If a passenger gets on a 16. horse located at the edge of the wheel and ravels 38 feet, find the angle of rotation to the nearest degree. ARL Sub and divide<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n\n<p>(5) Gina Wilson (All Things Avgebrab, LLC), 2015-2<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/image-464-1012x1024.png\" alt=\"\" class=\"wp-image-234251\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Of course. Here are the correct answers and a detailed, textbook-style explanation for each problem on the worksheet.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Conceptual Foundation: Arc Length and Central Angles<\/strong><\/h3>\n\n\n\n<p>The problems on this worksheet involve calculating the length of an arc of a circle. An&nbsp;<strong>arc<\/strong>&nbsp;is a portion of the circumference of a circle. Its length is directly proportional to the measure of its corresponding&nbsp;<strong>central angle<\/strong>\u2014the angle whose vertex is the center of the circle and whose sides are radii intersecting the circle at the arc&#8217;s endpoints.<\/p>\n\n\n\n<p>The formula to find the length of an arc is derived from the formula for the circumference of a full circle (C = 2\u03c0r). The arc length is a fraction of the total circumference, determined by the fraction of the full 360\u00b0 that the central angle represents.<\/p>\n\n\n\n<p>The governing formula is:<br><strong>Arc Length = (\u03b8 \/ 360\u00b0) \u00d7 2\u03c0r<\/strong><br>Where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u03b8<\/strong>\u00a0(theta) is the measure of the central angle in degrees.<\/li>\n\n\n\n<li><strong>r<\/strong>\u00a0is the radius of the circle.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Problem Solutions<\/strong><\/h3>\n\n\n\n<p><strong>Question 13<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Given:<\/strong>\u00a0Central angle (\u03b8) = 232\u00b0, Radius (r) = 8 mm.<\/li>\n\n\n\n<li><strong>Objective:<\/strong>\u00a0Find the length of the bolded arc.<\/li>\n<\/ul>\n\n\n\n<p>The bolded arc corresponds directly to the given central angle. Applying the arc length formula:<\/p>\n\n\n\n<p>Arc Length = (232\u00b0 \/ 360\u00b0) \u00d7 2\u03c0(8)<br>Arc Length = (232 \/ 360) \u00d7 16\u03c0<br>Arc Length \u2248 0.6444 \u00d7 16\u03c0<br>Arc Length \u2248 10.3111\u03c0<br>Arc Length \u2248 32.394&#8230; mm<\/p>\n\n\n\n<p>Rounding to the nearest hundredth, the length of the arc is&nbsp;<strong>32.39 mm<\/strong>.<\/p>\n\n\n\n<p><strong>Question 14<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Given:<\/strong>\u00a0Radius (r) = 14.5 ft. The un-bolded arc has a central angle of 60\u00b0.<\/li>\n\n\n\n<li><strong>Objective:<\/strong>\u00a0Find the length of the bolded (major) arc.<\/li>\n<\/ul>\n\n\n\n<p>First, determine the central angle of the bolded arc. A full circle is 360\u00b0.<br>\u03b8 = 360\u00b0 &#8211; 60\u00b0 = 300\u00b0<\/p>\n\n\n\n<p>Now, apply the arc length formula with this angle:<br>Arc Length = (300\u00b0 \/ 360\u00b0) \u00d7 2\u03c0(14.5)<br>Arc Length = (5 \/ 6) \u00d7 29\u03c0<br>Arc Length = (145\u03c0 \/ 6)<br>Arc Length \u2248 75.9218&#8230; ft<\/p>\n\n\n\n<p>Rounding to the nearest hundredth, the length of the arc is&nbsp;<strong>75.92 ft<\/strong>.<\/p>\n\n\n\n<p><strong>Question 15<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Given:<\/strong>\u00a0Diameter PS = 28 feet, which means the radius (r) is 14 feet. The measure of arc QR is 92\u00b0, and the measure of arc PT is 125\u00b0.<\/li>\n\n\n\n<li><strong>Objective:<\/strong>\u00a0Find the lengths of arc ST and arc RPT.<\/li>\n<\/ul>\n\n\n\n<p><strong>a) Find the length of arc ST.<\/strong><br>Since PS is a diameter, the arc from P through T to S (arc PTS) is a semicircle, which measures 180\u00b0.<br>Measure of arc ST = Measure of arc PTS &#8211; Measure of arc PT<br>Measure of arc ST = 180\u00b0 &#8211; 125\u00b0 = 55\u00b0<\/p>\n\n\n\n<p>Now, calculate the length of arc ST using its angle measure:<br>Arc Length (ST) = (55\u00b0 \/ 360\u00b0) \u00d7 2\u03c0(14)<br>Arc Length (ST) = (55 \/ 360) \u00d7 28\u03c0<br>Arc Length (ST) \u2248 0.1527 \u00d7 28\u03c0<br>Arc Length (ST) \u2248 13.439&#8230; ft<\/p>\n\n\n\n<p>Rounding to the nearest hundredth, the length of arc ST is&nbsp;<strong>13.44 ft<\/strong>.<\/p>\n\n\n\n<p><strong>b) Find the length of arc RPT.<\/strong><br>The arc RPT is the arc that travels from point R, through point P, to point T. Its measure is the sum of m(arc RP) + m(arc PT). We know m(arc PT) = 125\u00b0, but m(arc RP) is unknown. For the given information to be consistent, the segment RT must also be a diameter. This would make arc RPT a semicircle measuring 180\u00b0. (This is consistent because it would imply m(arc TQR) is also a semicircle, and m(arc TQ) + m(arc QR) = 88\u00b0 + 92\u00b0 = 180\u00b0).<\/p>\n\n\n\n<p>Therefore, the measure of arc RPT is 180\u00b0. Its length is half the circle&#8217;s circumference.<br>Arc Length (RPT) = (180\u00b0 \/ 360\u00b0) \u00d7 2\u03c0(14)<br>Arc Length (RPT) = (1\/2) \u00d7 28\u03c0<br>Arc Length (RPT) = 14\u03c0<br>Arc Length (RPT) \u2248 43.982&#8230; ft<\/p>\n\n\n\n<p>Rounding to the nearest hundredth, the length of arc RPT is&nbsp;<strong>43.98 ft<\/strong>.<\/p>\n\n\n\n<p><strong>Question 16<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Given:<\/strong>\u00a0Radius (r) = 20 feet, Arc Length = 38 feet.<\/li>\n\n\n\n<li><strong>Objective:<\/strong>\u00a0Find the angle of rotation (the central angle \u03b8) to the nearest degree.<\/li>\n<\/ul>\n\n\n\n<p>This problem requires rearranging the arc length formula to solve for the angle, \u03b8.<br>Arc Length = (\u03b8 \/ 360\u00b0) \u00d7 2\u03c0r<br>38 = (\u03b8 \/ 360) \u00d7 2\u03c0(20)<br>38 = (\u03b8 \/ 360) \u00d7 40\u03c0<\/p>\n\n\n\n<p>To isolate \u03b8, multiply both sides by 360 and divide by 40\u03c0:<br>38 \u00d7 360 = \u03b8 \u00d7 40\u03c0<br>13680 = \u03b8 \u00d7 40\u03c0<br>\u03b8 = 13680 \/ (40\u03c0)<br>\u03b8 = 342 \/ \u03c0<br>\u03b8 \u2248 108.86&#8230;\u00b0<\/p>\n\n\n\n<p>Rounding to the nearest degree, the angle of rotation is&nbsp;<strong>109\u00b0<\/strong>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-505.jpeg\" alt=\"\" class=\"wp-image-234252\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>For questions 13 and 14, find the length of each bolded arc. Round to the nearest hundredth. 13. 14. (5) Gina Wilson (All Things Avgebrab, LLC), 2015-2 The Correct Answer and Explanation is: Of course. Here are the correct answers and a detailed, textbook-style explanation for each problem on the worksheet. Conceptual Foundation: Arc Length [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-234250","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234250","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=234250"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234250\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=234250"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=234250"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=234250"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}