{"id":196148,"date":"2025-03-03T05:10:52","date_gmt":"2025-03-03T05:10:52","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=196148"},"modified":"2025-03-03T05:10:55","modified_gmt":"2025-03-03T05:10:55","slug":"determine-centroid-x-and-y-of-the-shaded-area-shown","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/03\/determine-centroid-x-and-y-of-the-shaded-area-shown\/","title":{"rendered":"Determine centroid X and Y of the shaded area shown"},"content":{"rendered":"\n<p>Determine centroid X and Y of the shaded area shown.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/03\/image-105.png\" alt=\"\" class=\"wp-image-196149\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To determine the centroid (center of mass) of the shaded area shown in the provided image, we can apply the principles of composite areas. The dimensions given are 2 ft \u00d7 2 ft, 3 ft \u00d7 1 ft, and 2 ft \u00d7 3 ft. \ue200cite\ue202turn0search5\ue201<\/p>\n\n\n\n<p><strong>Step 1: Break the Shape into Simpler Components<\/strong><\/p>\n\n\n\n<p>The shaded area can be decomposed into three rectangles:\ue206<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Rectangle A<\/strong>: Dimensions 2 ft \u00d7 2 ft\ue206<\/li>\n\n\n\n<li><strong>Rectangle B<\/strong>: Dimensions 3 ft \u00d7 1 ft\ue206<\/li>\n\n\n\n<li><strong>Rectangle C<\/strong>: Dimensions 2 ft \u00d7 3 ft\ue206<\/li>\n<\/ol>\n\n\n\n<p><strong>Step 2: Calculate the Area and Centroid of Each Rectangle<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Rectangle A<\/strong>:<\/li>\n\n\n\n<li>Area (A\u2081): 2 ft \u00d7 2 ft = 4 ft\u00b2\ue206<\/li>\n\n\n\n<li>Centroid coordinates (x\u2081, y\u2081): (1 ft, 1 ft)\ue206\n<ul class=\"wp-block-list\">\n<li>Located at half the length and half the width.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Rectangle B<\/strong>:<\/li>\n\n\n\n<li>Area (A\u2082): 3 ft \u00d7 1 ft = 3 ft\u00b2\ue206<\/li>\n\n\n\n<li>Centroid coordinates (x\u2082, y\u2082): (4.5 ft, 0.5 ft)\ue206\n<ul class=\"wp-block-list\">\n<li>Placed 1.5 ft from the left edge and 0.5 ft from the bottom edge.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Rectangle C<\/strong>:<\/li>\n\n\n\n<li>Area (A\u2083): 2 ft \u00d7 3 ft = 6 ft\u00b2\ue206<\/li>\n\n\n\n<li>Centroid coordinates (x\u2083, y\u2083): (1 ft, 4 ft)\ue206\n<ul class=\"wp-block-list\">\n<li>Situated 1 ft from the left edge and 4 ft from the bottom edge.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p><strong>Step 3: Compute the Weighted Average of Centroids<\/strong><\/p>\n\n\n\n<p>The overall centroid (X, Y) of the composite shape is found by calculating the weighted average of the centroids of the individual rectangles, weighted by their respective areas:\ue206<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Total Area (A)<\/strong>:<\/li>\n\n\n\n<li>A = A\u2081 + A\u2082 + A\u2083 = 4 ft\u00b2 + 3 ft\u00b2 + 6 ft\u00b2 = 13 ft\u00b2\ue206<\/li>\n\n\n\n<li><strong>X-coordinate of Centroid (X)<\/strong>:<\/li>\n\n\n\n<li>X = (A\u2081 \u00d7 x\u2081 + A\u2082 \u00d7 x\u2082 + A\u2083 \u00d7 x\u2083) \/ A\ue206<\/li>\n\n\n\n<li>X = (4 ft\u00b2 \u00d7 1 ft + 3 ft\u00b2 \u00d7 4.5 ft + 6 ft\u00b2 \u00d7 1 ft) \/ 13 ft\u00b2\ue206<\/li>\n\n\n\n<li>X = (4 ft\u00b2 + 13.5 ft\u00b2 + 6 ft\u00b2) \/ 13 ft\u00b2\ue206<\/li>\n\n\n\n<li>X = 23.5 ft\u00b2 \/ 13 ft\u00b2\ue206<\/li>\n\n\n\n<li>X \u2248 1.81 ft\ue206<\/li>\n\n\n\n<li><strong>Y-coordinate of Centroid (Y)<\/strong>:<\/li>\n\n\n\n<li>Y = (A\u2081 \u00d7 y\u2081 + A\u2082 \u00d7 y\u2082 + A\u2083 \u00d7 y\u2083) \/ A\ue206<\/li>\n\n\n\n<li>Y = (4 ft\u00b2 \u00d7 1 ft + 3 ft\u00b2 \u00d7 0.5 ft + 6 ft\u00b2 \u00d7 4 ft) \/ 13 ft\u00b2\ue206<\/li>\n\n\n\n<li>Y = (4 ft\u00b2 + 1.5 ft\u00b2 + 24 ft\u00b2) \/ 13 ft\u00b2\ue206<\/li>\n\n\n\n<li>Y = 29.5 ft\u00b2 \/ 13 ft\u00b2\ue206<\/li>\n\n\n\n<li>Y \u2248 2.27 ft\ue206<\/li>\n<\/ul>\n\n\n\n<p><strong>Conclusion<\/strong><\/p>\n\n\n\n<p>The centroid of the shaded area is located approximately at coordinates (X, Y) = (1.81 ft, 2.27 ft) relative to the bottom-left corner of the shape.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Determine centroid X and Y of the shaded area shown. The Correct Answer and Explanation is : To determine the centroid (center of mass) of the shaded area shown in the provided image, we can apply the principles of composite areas. The dimensions given are 2 ft \u00d7 2 ft, 3 ft \u00d7 1 ft, [&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-196148","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196148","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=196148"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196148\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=196148"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=196148"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=196148"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}