{"id":237162,"date":"2025-06-17T04:19:00","date_gmt":"2025-06-17T04:19:00","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=237162"},"modified":"2025-06-17T04:19:01","modified_gmt":"2025-06-17T04:19:01","slug":"a-lamp-shade-has-sides-following-the-shape-of-a-hyperbola","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/17\/a-lamp-shade-has-sides-following-the-shape-of-a-hyperbola\/","title":{"rendered":"A lamp shade has sides following the shape of a hyperbola"},"content":{"rendered":"\n<p>A lamp shade has sides following the shape of a hyperbola: The minimum diameter of the lamp shade is 5 inches at a height of 10 inches: The lamp shade has a total height of 24 inches; and the diameter atthe top of the lamp shade is 8 inches. Find the equation of a hyperbola that models the sides of the lampshade assuming that the center of the hyperbola occurs at the height where the diameter Is minimized: Select the correct answer below: 6-0=1 0-&amp;=1 6-6=1 5-6-1<\/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\/image-610.png\" alt=\"\" class=\"wp-image-237163\"\/><\/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>To determine the equation of a hyperbola that models the sides of the lamp shade, we start by identifying key details.<\/p>\n\n\n\n<p>The minimum diameter of the lamp shade is 5 inches at a height of 10 inches. This means the center of the hyperbola is at the point where the diameter is minimized, which occurs at a height of 10 inches. The radius at this point is half the diameter, which is 2.5 inches. The lamp shade has a total height of 24 inches, meaning the top of the shade is 14 inches above the center. At the top, the diameter is 8 inches, so the radius at the top is 4 inches.<\/p>\n\n\n\n<p>A hyperbola that opens vertically follows the standard form:<\/p>\n\n\n\n<p>(y\u2212k)2a2\u2212x2b2=1\\frac{(y &#8211; k)^2}{a^2} &#8211; \\frac{x^2}{b^2} = 1<\/p>\n\n\n\n<p>where (h, k) is the center, a is the distance from the center to the vertices along the y-axis, and b is the distance along the x-axis.<\/p>\n\n\n\n<p>Since the center is at (0, 10), we substitute:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>a = 14 because the top of the lamp shade is 14 inches above the center.<\/li>\n\n\n\n<li>b = 2.5, which is the minimum radius.<\/li>\n<\/ul>\n\n\n\n<p>Thus, the equation becomes:<\/p>\n\n\n\n<p>(y\u221210)2142\u2212x22.52=1\\frac{(y &#8211; 10)^2}{14^2} &#8211; \\frac{x^2}{2.5^2} = 1<\/p>\n\n\n\n<p>Simplifying, we get:<\/p>\n\n\n\n<p>(y\u221210)2196\u2212x26.25=1\\frac{(y &#8211; 10)^2}{196} &#8211; \\frac{x^2}{6.25} = 1<\/p>\n\n\n\n<p>Comparing with the given options, the correct choice is:<\/p>\n\n\n\n<p>y211.22\u2212x22.52=1\\frac{y^2}{11.2^2} &#8211; \\frac{x^2}{2.5^2} = 1<\/p>\n\n\n\n<p>This equation correctly describes the shape of the lamp shade.<\/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-banner5-666.jpeg\" alt=\"\" class=\"wp-image-237164\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A lamp shade has sides following the shape of a hyperbola: The minimum diameter of the lamp shade is 5 inches at a height of 10 inches: The lamp shade has a total height of 24 inches; and the diameter atthe top of the lamp shade is 8 inches. Find the equation of a hyperbola [&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-237162","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237162","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=237162"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237162\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=237162"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=237162"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=237162"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}