{"id":43491,"date":"2025-06-30T06:50:00","date_gmt":"2025-06-30T06:50:00","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=43491"},"modified":"2025-06-30T06:50:01","modified_gmt":"2025-06-30T06:50:01","slug":"two-cones-have-their-heights-in-the-ratio-13-and-radii-in-the-ratio-3-1","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/two-cones-have-their-heights-in-the-ratio-13-and-radii-in-the-ratio-3-1\/","title":{"rendered":"Two cones have their heights in the ratio 1:3 and radii in the ratio 3: 1"},"content":{"rendered":"\n<p>Two cones have their heights in the ratio 1:3 and radii in the ratio 3: 1. What is the ratio of their volumes?<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To find the ratio of the volumes of two cones, we use the formula for the volume of a cone: V=13\u03c0r2hV = \\frac{1}{3} \\pi r^2 h<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>rr is the radius,<\/li>\n\n\n\n<li>hh is the height, and<\/li>\n\n\n\n<li>VV is the volume of the cone.<\/li>\n<\/ul>\n\n\n\n<p>Let the height of the first cone be h1h_1 and the radius be r1r_1. Let the height of the second cone be h2h_2 and the radius be r2r_2. We are given the ratio of heights as: h1h2=13\\frac{h_1}{h_2} = \\frac{1}{3}<\/p>\n\n\n\n<p>and the ratio of radii as: r1r2=31\\frac{r_1}{r_2} = \\frac{3}{1}<\/p>\n\n\n\n<p>The volume of the first cone is: V1=13\u03c0r12h1V_1 = \\frac{1}{3} \\pi r_1^2 h_1<\/p>\n\n\n\n<p>The volume of the second cone is: V2=13\u03c0r22h2V_2 = \\frac{1}{3} \\pi r_2^2 h_2<\/p>\n\n\n\n<p>To find the ratio of the volumes, we divide V1V_1 by V2V_2: V1V2=13\u03c0r12h113\u03c0r22h2\\frac{V_1}{V_2} = \\frac{\\frac{1}{3} \\pi r_1^2 h_1}{\\frac{1}{3} \\pi r_2^2 h_2}<\/p>\n\n\n\n<p>The 13\u03c0\\frac{1}{3} \\pi terms cancel out, leaving: V1V2=r12h1r22h2\\frac{V_1}{V_2} = \\frac{r_1^2 h_1}{r_2^2 h_2}<\/p>\n\n\n\n<p>Substituting the given ratios r1r2=3\\frac{r_1}{r_2} = 3 and h1h2=13\\frac{h_1}{h_2} = \\frac{1}{3}, we get: V1V2=(3)2\u00d71(1)2\u00d73=93=3\\frac{V_1}{V_2} = \\frac{(3)^2 \\times 1}{(1)^2 \\times 3} = \\frac{9}{3} = 3<\/p>\n\n\n\n<p>Therefore, the ratio of the volumes of the two cones is: 3:13:1<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation:<\/h3>\n\n\n\n<p>The volume of a cone is directly proportional to the square of the radius and the height. When the ratio of the radii is 3:1, the square of the radius ratio becomes 32=93^2 = 9. Since the ratio of the heights is 1:3, the height ratio contributes a factor of 13\\frac{1}{3}. Combining these factors, we get a final ratio of 9\u00d713=39 \\times \\frac{1}{3} = 3, so the volume ratio is 3:1.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Two cones have their heights in the ratio 1:3 and radii in the ratio 3: 1. What is the ratio of their volumes? The correct answer and explanation is: To find the ratio of the volumes of two cones, we use the formula for the volume of a cone: V=13\u03c0r2hV = \\frac{1}{3} \\pi r^2 h [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-43491","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/43491","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=43491"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/43491\/revisions"}],"predecessor-version":[{"id":43492,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/43491\/revisions\/43492"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=43491"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=43491"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=43491"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}