Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 5.<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n The correct answer is: {\\frac{125}{6}} \\text{ cubic units}<\/strong><\/mark><\/p>\n\n\n\n To find the volume of the solid bounded by the planes (x = 0), (y = 0), (z = 0), and (x + y + z = 5), we can visualize this solid as a tetrahedron in the first octant of three-dimensional space. The equation (x + y + z = 5) represents a plane that intersects the axes at the points where (x), (y), or (z) equals 5.<\/p>\n\n\n\n The volume of the solid bounded by the given planes is: This result indicates that the solid occupies approximately (20.83) cubic units. Understanding this volume in relation to the first octant and how the tetrahedron is formed helps visualize the three-dimensional shape and the intersection of the planes. The tetrahedral shape is a fundamental geometric concept in calculus and can be extended to higher dimensions and more complex shapes.<\/p>\n","protected":false},"excerpt":{"rendered":" Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 5. The Correct Answer and Explanation is : The correct answer is: {\\frac{125}{6}} \\text{ cubic units} To find the volume of the solid bounded by the planes (x = […]<\/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-149768","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/149768","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=149768"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/149768\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=149768"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=149768"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=149768"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Steps to Determine the Volume:<\/h3>\n\n\n\n
\n
\n
\n
\n
The volume (V) of a tetrahedron with vertices at the origin and points ( (a, 0, 0) ), ( (0, b, 0) ), ( (0, 0, c) ) is given by the formula:
[
V = \\frac{1}{6} \\times a \\times b \\times c
]<\/li>\n\n\n\n
Here, (a = 5), (b = 5), and (c = 5):
[
V = \\frac{1}{6} \\times 5 \\times 5 \\times 5 = \\frac{125}{6}
]<\/li>\n<\/ol>\n\n\n\nConclusion:<\/h3>\n\n\n\n
[
\\boxed{\\frac{125}{6}} \\text{ cubic units}
]<\/p>\n\n\n\n