{"id":9917,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9917"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"surface-area-and-volume-of-solids-conversion-of-solids-frustum-of-a-cone","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/surface-area-and-volume-of-solids-conversion-of-solids-frustum-of-a-cone\/","title":{"rendered":"Surface Area And Volume Of Solids, Conversion Of Solids, Frustum Of A Cone"},"content":{"rendered":"<div class=\"article-watermark-wrapper\">\n<div style=\"position: relative; z-index: 1;\">\n<p style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 9pt; color: #444444;\">KAPDEC&reg; | Elite STEM Learning Platform | <a href=\"https:\/\/kapdec.com\" target=\"_blank\" rel=\"noopener noreferrer\" style=\"color: #444444; text-decoration: underline;\">https:\/\/kapdec.com<\/a><\/p>\n<hr \/>\n<h2><strong>Unit: <\/strong><strong>Measurement System<\/strong><\/h2>\n<h3><strong>Chapter: <\/strong><strong>Surface Area and Volume of Solids, Conversion of Solids, Frustum of a Cone<\/strong><\/h3>\n<p><em>Reference: &#8211; Introduction to Solids, Surface Area and Volume of Cuboid, Cube, Cylinder, Cone, Sphere, Conversion of Solids from One Shape to Another, Frustum of a Cone &#8211; Surface Area and Volume, Applications and Word Problems<\/em><\/p>\n<p><strong>After studying this chapter, you should be able to understand:<\/strong><\/p>\n<ul>\n<li>The concepts of surface area and volume for common 3D shapes.<\/li>\n<li>How to calculate the surface area and volume of a frustum of a cone.<\/li>\n<li>The principle of conversion of solids and its applications.<\/li>\n<li>How to solve real-world problems involving these concepts.<\/li>\n<\/ul>\n<p><strong>Introduction to Solids<\/strong><\/p>\n<p><strong><u>Definition<\/u><\/strong><\/p>\n<p>A solid is a three-dimensional object that has length, breadth, and height (or depth). Unlike 2D shapes, solids occupy space and have volume. Common solids include cubes, cuboids, cylinders, cones, and spheres.<\/p>\n<p>The surface area is the total area of the outer surfaces of the solid, while the volume is the amount of space enclosed by the solid.<\/p>\n<p><strong>[Importance of Solids]<\/strong><\/p>\n<ul>\n<li>Essential for understanding objects in the real world, from boxes to buildings.<\/li>\n<li>Used in fields like architecture, engineering, and manufacturing.<\/li>\n<li>Helps in calculating material requirements, capacity, and cost.<\/li>\n<li>Forms the basis for more advanced topics in mathematics and physics.<\/li>\n<\/ul>\n<p><strong>Example<\/strong><\/p>\n<p><strong>A cardboard box<\/strong>\u00a0is an example of a cuboid. Its surface area would be the area of cardboard used, and its volume would be the space inside it.<\/p>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Types of Solids<\/strong><\/p>\n<ul>\n<li><strong>Polyhedra:<\/strong>\u00a0Solids with flat faces (e.g., cube, cuboid, pyramid).<\/li>\n<li><strong>Curved Solids:<\/strong>\u00a0Solids with curved surfaces (e.g., cylinder, cone, sphere).<\/li>\n<\/ul>\n<p><strong>Key Points:<\/strong><\/p>\n<ul>\n<li><strong>Lateral Surface Area (LSA):<\/strong>\u00a0The area of all faces excluding the top and bottom.<\/li>\n<li><strong>Total Surface Area (TSA):<\/strong>\u00a0The area of all faces, including top and bottom.<\/li>\n<li><strong>Volume:<\/strong>\u00a0The measure of the space occupied by the solid.<\/li>\n<\/ul>\n<p><strong>Surface Area and Volume of Cuboid and Cube<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<ul>\n<li>A\u00a0<strong>cuboid<\/strong>\u00a0is a solid with six rectangular faces. It has length (l), breadth (b), and height (h).<\/li>\n<li>A\u00a0<strong>cube<\/strong>\u00a0is a special cuboid where length = breadth = height = a.<\/li>\n<\/ul>\n<p><strong>[Importance of Cuboid and Cube]<\/strong><\/p>\n<ul>\n<li>Most common shapes for packaging and storage.<\/li>\n<li>Easy to model and calculate for various applications.<\/li>\n<li>Foundation for understanding more complex solids.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>Find the TSA and volume of a cuboid with l=5 cm, b=4 cm, h=3 cm.<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Formulas for Cuboid<\/strong><\/p>\n<ul>\n<li><strong>Volume (V) = l \u00d7 b \u00d7 h<\/strong><\/li>\n<li><strong>LSA = 2h(l + b)<\/strong><\/li>\n<li><strong>TSA = 2(lb + bh + hl)<\/strong><\/li>\n<\/ul>\n<p><strong>2. Formulas for Cube<\/strong><\/p>\n<ul>\n<li><strong>Volume (V) = a\u00b3<\/strong><\/li>\n<li><strong>LSA = 4a\u00b2<\/strong><\/li>\n<li><strong>TSA = 6a\u00b2<\/strong><\/li>\n<\/ul>\n<p><strong>Surface Area and Volume of Cylinder<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>A cylinder is a solid with two parallel circular bases connected by a curved surface. It has a height (h) and a base radius (r).<\/p>\n<p><strong>[Importance of Cylinder]<\/strong><\/p>\n<ul>\n<li>Used in containers like cans, pipes, and tanks.<\/li>\n<li>Common in mechanical and civil engineering.<\/li>\n<li>Helps in understanding curved surface areas.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>Find the volume of a cylinder with r=7 cm and h=10 cm.<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Formulas for Cylinder<\/strong><\/p>\n<ul>\n<li><strong>Volume (V) = \u03c0r\u00b2h<\/strong><\/li>\n<li><strong>Curved Surface Area (CSA) = 2\u03c0rh<\/strong><\/li>\n<li><strong>Total Surface Area (TSA) = 2\u03c0r(h + r)<\/strong><\/li>\n<\/ul>\n<p><strong>Surface Area and Volume of Cone<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>A cone is a solid that tapers smoothly from a flat circular base to a point called the apex or vertex. It has a base radius (r), height (h), and slant height (l).<\/p>\n<p><strong>[Importance of Cone]<\/strong><\/p>\n<ul>\n<li>Used in funnels, ice cream cones, and party hats.<\/li>\n<li>Important in geometry and calculus.<\/li>\n<li>Helps in understanding the concept of slant height.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>Find the slant height of a cone with r=3 cm and h=4 cm.<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Formulas for Cone<\/strong><\/p>\n<ul>\n<li><strong>Slant Height (l) = \u221a(r\u00b2 + h\u00b2)<\/strong><\/li>\n<li><strong>Volume (V) = (1\/3)\u03c0r\u00b2h<\/strong><\/li>\n<li><strong>CSA = \u03c0rl<\/strong><\/li>\n<li><strong>TSA = \u03c0r(l + r)<\/strong><\/li>\n<\/ul>\n<p><strong>Surface Area and Volume of Sphere<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>A sphere is a perfectly round geometrical object in three-dimensional space, like a ball. It is defined by its radius (r).<\/p>\n<p><strong>[Importance of Sphere]<\/strong><\/p>\n<ul>\n<li>Models objects like planets, balls, and bubbles.<\/li>\n<li>Used in physics and astronomy.<\/li>\n<li>Has the smallest surface area for a given volume.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>Find the surface area of a sphere with r=7 cm.<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Formulas for Sphere<\/strong><\/p>\n<ul>\n<li><strong>Volume (V) = (4\/3)\u03c0r\u00b3<\/strong><\/li>\n<li><strong>Surface Area (SA) = 4\u03c0r\u00b2<\/strong><\/li>\n<\/ul>\n<p><strong>Conversion of Solids from One Shape to Another<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>This concept involves melting or reshaping a solid into another solid without any loss of material. The volume remains constant during conversion.<\/p>\n<p><strong>[Importance of Conversion]<\/strong><\/p>\n<ul>\n<li>Practical in metallurgy and manufacturing.<\/li>\n<li>Helps in solving problems involving material reuse.<\/li>\n<li>Tests understanding of volume conservation.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>A metallic sphere of radius 6 cm is melted and recast into a cylinder of radius 3 cm. Find the height of the cylinder.<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Principle<\/strong><\/p>\n<p>Volume of original solid = Volume of new solid<\/p>\n<p><strong>2. Application<\/strong><\/p>\n<p>Set up an equation using the volume formulas of both solids and solve for the unknown dimension.<\/p>\n<p><strong>Frustum of a Cone<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>When a cone is cut by a plane parallel to its base, the portion between the base and the cutting plane is called a frustum of the cone. It has two circular bases of different radii.<\/p>\n<p><strong>[Importance of Frustum]<\/strong><\/p>\n<ul>\n<li>Common in buckets, lampshades, and certain architectural elements.<\/li>\n<li>Extends the understanding of cones to truncated shapes.<\/li>\n<li>Useful in practical volume and surface area calculations.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>Find the volume of a frustum with radii 3 cm and 5 cm, and height 6 cm.<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Elements of a Frustum<\/strong><\/p>\n<ul>\n<li><strong>R:<\/strong>\u00a0Radius of the larger base.<\/li>\n<li><strong>r:<\/strong>\u00a0Radius of the smaller base.<\/li>\n<li><strong>h:<\/strong>\u00a0Height of the frustum (vertical distance between bases).<\/li>\n<li><strong>l:<\/strong>\u00a0Slant height of the frustum,\u00a0\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"33\" src=\"https:\/\/app.kapdec.com\/questions-images\/98fsG5yJVd4z1765107990.gif?time=1765107991\" width=\"165\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>.<\/li>\n<\/ul>\n<p><strong>2. Formulas for Frustum<\/strong><\/p>\n<ul>\n<li><strong>Volume (V) = (1\/3)\u03c0h (R\u00b2 + r\u00b2 + Rr)<\/strong><\/li>\n<li><strong>CSA = \u03c0l (R + r)<\/strong><\/li>\n<li><strong>TSA = CSA + \u03c0(R\u00b2 + r\u00b2)<\/strong><\/li>\n<\/ul>\n<p><strong>Applications and Word Problems<\/strong><\/p>\n<p><strong>[Definition]<\/strong><\/p>\n<p>These problems involve applying the formulas for surface area, volume, and conversion to real-life situations. They often require multiple steps and logical reasoning.<\/p>\n<p><strong>[Importance of Word Problems]<\/strong><\/p>\n<ul>\n<li>Bridges theoretical math with practical application.<\/li>\n<li>Enhances problem-solving and analytical skills.<\/li>\n<li>Common in academic and competitive exams.<\/li>\n<\/ul>\n<p><strong>Examples<\/strong><\/p>\n<ul>\n<li>A tent is in the shape of a cylinder surmounted by a cone. Find the canvas required for the tent.<\/li>\n<\/ul>\n<p><strong>[Subtopics]<\/strong><\/p>\n<p><strong>1. Problem-Solving Strategy<\/strong><\/p>\n<ol>\n<li>Understand the problem and identify the solids involved.<\/li>\n<li>Note down the given dimensions.<\/li>\n<li>Determine which formulas are needed (SA, Volume, etc.).<\/li>\n<li>Perform the calculations step by step.<\/li>\n<li>Ensure units are consistent and interpret the result.<\/li>\n<\/ol>\n<p><strong>[Example: -]<\/strong><\/p>\n<p><strong>Problem Statement:<\/strong><br \/>\nA solid metallic sphere of radius 10.5 cm is melted and recast into a number of smaller cones, each of radius 3.5 cm and height 3 cm. Find the number of cones formed.<br \/>\nAlso, a bucket (frustum of a cone) has top and bottom radii of 28 cm and 21 cm respectively, and a height of 15 cm. Find its capacity in liters.<\/p>\n<p><strong>Question:<\/strong>\u00a0Solve both parts. Prove your answers by providing a step-by-step solution and giving\u00a0<strong>three independent reasons<\/strong>\u00a0supporting your conclusion for the first part from these domains:\u00a0<strong>(A) Volume Conservation Principle, (B) Mathematical Calculation, (C) Logical Unit Analysis.<\/strong><\/p>\n<p><strong>[Solution: -]<\/strong><\/p>\n<p><strong>Part 1: Number of Cones Formed<\/strong><\/p>\n<p><strong>Given:<\/strong><\/p>\n<ul>\n<li>Radius of sphere,\u00a0\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/cq7CdEUnoTfJ1765107991.gif?time=1765107991\" width=\"84\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0cm<\/li>\n<li>Radius of each cone,\u00a0\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/ppS1XhMW5Z731765107991.gif?time=1765107991\" width=\"67\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0cm<\/li>\n<li>Height of each cone,\u00a0\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/87MxHecSVTn61765107991.gif?time=1765107991\" width=\"57\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0cm<\/li>\n<\/ul>\n<p><strong>(A) Volume Conservation Principle<\/strong><br \/>\nWhen a solid is melted and recast, its volume remains unchanged.<br \/>\nTherefore, Volume of Sphere = Number of cones \u00d7 Volume of one cone.<\/p>\n<p><strong>(B) Mathematical Calculation<\/strong><br \/>\nFirst, calculate the volume of the sphere:<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"49\" src=\"https:\/\/app.kapdec.com\/questions-images\/UJNlomksktAn1765107991.gif?time=1765107992\" width=\"302\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>Compute\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/FOLR38X9HgoO1765107992.gif?time=1765107992\" width=\"355\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"49\" src=\"https:\/\/app.kapdec.com\/questions-images\/Aq3E1pnn8XLn1765107992.gif?time=1765107992\" width=\"246\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>Simplify step-by-step:<br \/>\nFirst,\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"37\" src=\"https:\/\/app.kapdec.com\/questions-images\/BWsHvn77IAPT1765107992.gif?time=1765107992\" width=\"199\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\nThen,\u00a0..<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"37\" src=\"https:\/\/app.kapdec.com\/questions-images\/yZvYsu3IZfes1765107992.gif?time=1765107992\" width=\"305\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\nLet&#8217;s compute precisely:<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/zPMCDEsGHbab1765107992.gif?time=1765107993\" width=\"186\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>Then,\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/A9y8RHvIWn1w1765107993.gif?time=1765107993\" width=\"147\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0cm\u00b3.<br \/>\nSo,\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"29\" src=\"https:\/\/app.kapdec.com\/questions-images\/PBktFEquOFjx1765107993.gif?time=1765107993\" width=\"129\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0cm\u00b3.<\/p>\n<p>Now, volume of one cone:<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"49\" src=\"https:\/\/app.kapdec.com\/questions-images\/i8SwJ40FFKGf1765107993.gif?time=1765107993\" width=\"328\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>Compute\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/YqDfLuIgwxuy1765107993.gif?time=1765107993\" width=\"126\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"49\" src=\"https:\/\/app.kapdec.com\/questions-images\/khauShd9bFaJ1765107993.gif?time=1765107994\" width=\"232\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>The 3 in numerator and denominator cancel:<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"49\" src=\"https:\/\/app.kapdec.com\/questions-images\/cCqgQcwSmZC01765107993.gif?time=1765107994\" width=\"164\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/R0LsQ9OxhtZT1765107994.gif?time=1765107994\" width=\"168\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><em>,\u00a0<\/em><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/WtOFp8NXDEZ81765107994.gif?time=1765107994\" width=\"133\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0cm\u00b3.<\/p>\n<p>Number of cones,\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"42\" src=\"https:\/\/app.kapdec.com\/questions-images\/AOLzaAilLjBk1765107994.gif?time=1765107994\" width=\"146\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>Compute:\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/R9wDyL335eCG1765107994.gif?time=1765107995\" width=\"529\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\nSo,\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/aWITuPdA2WeA1765107994.gif?time=1765107995\" width=\"71\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>.<\/p>\n<p><strong>(C) Logical Unit Analysis<\/strong><br \/>\nThe volumes are both in cm\u00b3, so the ratio<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"42\" src=\"https:\/\/app.kapdec.com\/questions-images\/0m7agiVBDA1N1765107995.gif?time=1765107995\" width=\"49\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0is a dimensionless number, correctly giving the number of cones. The calculation is consistent with unit analysis.<\/p>\n<p><strong>Therefore, the number of cones formed is 126.<\/strong><\/p>\n<p><strong>Part 2: Capacity of the Bucket (Frustum)<\/strong><\/p>\n<p><strong>Given:<\/strong><\/p>\n<ul>\n<li>Top radius,\u00a0<em>R=28<\/em>\u00a0cm<\/li>\n<li>Bottom radius,\u00a0<em>r=21<\/em>\u00a0cm<\/li>\n<li>Height,\u00a0<em>h=15<\/em>\u00a0cm<\/li>\n<\/ul>\n<p>The bucket is a frustum of a cone. Its capacity is its volume.<br \/>\nVolume of frustum:<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"49\" src=\"https:\/\/app.kapdec.com\/questions-images\/f22huiyyXW4G1765107995.gif?time=1765107996\" width=\"206\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>Substitute the values:<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"49\" src=\"https:\/\/app.kapdec.com\/questions-images\/sDPMUuu1KPwc1765107996.gif?time=1765107996\" width=\"358\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>Compute the terms inside:<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/tJ0on52HY6Bl1765107996.gif?time=1765107996\" width=\"90\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/wRIwWQhobT9s1765107996.gif?time=1765107996\" width=\"90\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/a9eVWdACezJA1765107996.gif?time=1765107997\" width=\"127\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>Sum =\u00a0<em>784+441+588=1813<\/em><\/p>\n<p>Now,<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"49\" src=\"https:\/\/app.kapdec.com\/questions-images\/nJ8PBa8cJ4pu1765107997.gif?time=1765107997\" width=\"208\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>First,\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"37\" src=\"https:\/\/app.kapdec.com\/questions-images\/LlKTecLqXYUO1765107997.gif?time=1765107997\" width=\"91\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\nSo,\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"37\" src=\"https:\/\/app.kapdec.com\/questions-images\/iOikOXPwEFi61765107997.gif?time=1765107997\" width=\"157\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"27\" src=\"https:\/\/app.kapdec.com\/questions-images\/luQozGNvCOhn1765107997.gif?time=1765107997\" width=\"171\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p><em>39886<\/em>\/<em>7=5698<\/em>\u00a0cm\u00b3 (approximately, let&#8217;s compute exactly).<\/p>\n<p>Actually,\u00a0<em>1813<\/em>\/<em>7=259<\/em>\u00a0exactly? Let&#8217;s check:\u00a0<em>7\u00d7259=1813<\/em>. Yes!<br \/>\nSo,\u00a0<em>V=5\u00d722\u00d7259=110\u00d7259=28490<\/em>\u00a0cm\u00b3.<\/p>\n<p>Now, convert to liters. Since 1 liter = 1000 cm\u00b3,<br \/>\nCapacity =\u00a0<\/p>\n<div class=\"kapdec-figure-wrapper\" style=\"display: inline-block; max-width: 100%; vertical-align: top;\"><img loading=\"lazy\" decoding=\"async\" alt=\"\" height=\"37\" src=\"https:\/\/app.kapdec.com\/questions-images\/iUHRb8zXm48s1765107998.gif?time=1765107998\" width=\"115\"><\/p>\n<p class=\"kapdec-figure-source\" style=\"font-family: Arial, Helvetica, Calibri, sans-serif; font-size: 8pt; color: #666666; text-align: right; margin: 4px 0 12px 0;\">Source: Kapdec.com<\/p>\n<\/div>\n<p>\u00a0liters \u2248\u00a0<strong>28.5 liters<\/strong>.<\/p>\n<p><strong>Final Answers:<\/strong><\/p>\n<ul>\n<li>Number of cones formed = 126<\/li>\n<li>Capacity of the bucket = 28.5 liters (approximately)<\/li>\n<\/ul>\n<p>The solution for the number of cones is verified by the principle of volume conservation, precise mathematical computation, and logical unit analysis.<\/p>\n<p>\u00a0<\/p>\n<p>\u00a0<\/p>\n<p><!--kapdec-footer-start--><\/p>\n<style>.kapdec-article-footer{font-family:Arial,Helvetica,Calibri,sans-serif;color:#444;}.kapdec-footer-grid{display:flex;align-items:stretch;border:1px solid #e5e7eb;border-radius:6px;overflow:hidden;}.kapdec-footer-left,.kapdec-qr-block{flex:1 1 50%;width:50%;box-sizing:border-box;min-width:0;}.kapdec-footer-left{padding:22px 28px;border-right:1px solid #e5e7eb;}.kapdec-citation-block{line-height:1.6;font-size:9pt;color:#333;margin:0;}.kapdec-citation-block p{margin:0 0 10px 0;}.kapdec-citation-block a{color:#0066cc;text-decoration:underline;}.kapdec-copyright-block{margin-top:18px;padding-top:14px;border-top:1px solid #e5e7eb;font-size:7.5pt;color:#777;line-height:1.55;text-align:left;}.kapdec-copyright-block p{margin:0 0 5px 0;}.kapdec-qr-block{padding:22px 28px;display:flex;flex-direction:column;align-items:center;justify-content:center;text-align:center;}.kapdec-qr-label{margin:0 0 8px 0;font-size:8.5pt;font-weight:600;color:#444;line-height:1.35;letter-spacing:.02em;}.kapdec-qr-url{margin:0 0 14px 0;font-size:7.5pt;line-height:1.4;color:#777;word-break:break-word;max-width:100%;}.kapdec-qr-url a{color:#777;text-decoration:underline;}@media (max-width:640px){.kapdec-footer-grid{flex-direction:column;}.kapdec-footer-left,.kapdec-qr-block{width:100%;flex-basis:100%;border-right:none;}.kapdec-footer-left{border-bottom:1px solid #e5e7eb;}}<\/style>\n<div class=\"kapdec-article-footer\" style=\"margin-top: 28px; padding-top: 4px;\">\n<div class=\"kapdec-footer-grid\">\n<div class=\"kapdec-footer-left\">\n<div class=\"kapdec-citation-block\">\n<p>A Kapdec&reg; learning guide &#8211; Crafted by elite STEM mentors for ambitious learners.<\/p>\n<p><a href=\"https:\/\/kapdec.com\" target=\"_blank\" rel=\"noopener noreferrer\">Learn more at https:\/\/kapdec.com<\/a><\/p>\n<\/div>\n<div class=\"kapdec-copyright-block\">\n<p>Author: Kapdec | Publisher: Kapdec | Copyright: &copy; Kapdec. 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