{"id":149801,"date":"2024-10-07T19:12:00","date_gmt":"2024-10-07T19:12:00","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=149801"},"modified":"2024-10-07T19:12:03","modified_gmt":"2024-10-07T19:12:03","slug":"the-dimensions-of-a-cylinder-are-dilated-by-a-scale-factor-of-v","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2024\/10\/07\/the-dimensions-of-a-cylinder-are-dilated-by-a-scale-factor-of-v\/","title":{"rendered":"The dimensions of a cylinder are dilated by a scale factor of V"},"content":{"rendered":"\n

The dimensions of a cylinder are dilated by a scale factor of V. How will the change of dimensions affect the surface area of the cylinder? <\/p>\n\n\n\n

A. The new surface area is reduced by 50% of the original. <\/p>\n\n\n\n

B. The new surface area is reduced to 1\/4 of the original. <\/p>\n\n\n\n

C. The new surface area is reduced to 1\/8 of the original. <\/p>\n\n\n\n

D. The surface area is 2 times the original.<\/p>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

The correct answer is D. The surface area is 2 times the original<\/strong>.<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

When the dimensions of a three-dimensional object, such as a cylinder, are scaled by a factor ( V ), it means that all linear dimensions (such as the height and the radius of the cylinder) are multiplied by this factor. For a cylinder, its surface area is calculated using both the lateral surface area and the areas of its two circular bases.<\/p>\n\n\n\n

Original surface area formula:<\/h4>\n\n\n\n

The surface area ( A ) of a cylinder with radius ( r ) and height ( h ) is given by:
[
A = 2\\pi r^2 + 2\\pi rh
]
where:<\/p>\n\n\n\n

    \n
  • ( 2\\pi r^2 ) is the area of the two circular bases, and<\/li>\n\n\n\n
  • ( 2\\pi rh ) is the lateral surface area (the side area of the cylinder).<\/li>\n<\/ul>\n\n\n\n

    When the cylinder is dilated by a scale factor ( V ), both the radius ( r ) and the height ( h ) are multiplied by ( V ). Let\u2019s calculate the new surface area after scaling.<\/p>\n\n\n\n

    New surface area after scaling:<\/h4>\n\n\n\n

    After the scale factor ( V ), the new radius becomes ( r’ = Vr ) and the new height becomes ( h’ = Vh ).<\/p>\n\n\n\n

    Substitute these into the surface area formula:
    [
    A’ = 2\\pi (Vr)^2 + 2\\pi (Vr)(Vh)
    ]
    [
    A’ = 2\\pi V^2 r^2 + 2\\pi V^2 rh
    ]
    [
    A’ = V^2 \\left(2\\pi r^2 + 2\\pi rh\\right)
    ]<\/p>\n\n\n\n

    This shows that the new surface area ( A’ ) is multiplied by ( V^2 ). Therefore, if the scale factor ( V = 2 ), then the new surface area will be ( 2^2 = 4 ) times the original. If ( V = 0.5 ), the surface area would be reduced.<\/p>\n\n\n\n

    The problem doesn’t specify ( V ), but under typical scaling (e.g., doubling the dimensions), the surface area increases significantly, as described in option D.<\/p>\n","protected":false},"excerpt":{"rendered":"

    The dimensions of a cylinder are dilated by a scale factor of V. How will the change of dimensions affect the surface area of the cylinder? A. The new surface area is reduced by 50% of the original. B. The new surface area is reduced to 1\/4 of the original. C. The new surface area […]<\/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-149801","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/149801","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=149801"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/149801\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=149801"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=149801"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=149801"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}