A cone of base radius 4 cm is divided into two parts by drawing a plane through the mid-point of its height and parallel to its base. Compare the volume of the two parts.

Given, radius of base of cone = 4 cm The cone is divided in two parts through the height of the cone. Let, the height of the cone is h cm . So, the volume of the cone = 1 3 × π × 4 2 × h cm 3 = 16 πh 3 cm 3 ∴ Height of the smaller cone is h 2 cm and radius is 4 2 cm = 2 cm ∴ Volume of smaller cone = 1 3 π × 2 2 × h 2 cm 3 = 2 πh 3 cm 3 ∴ Volume of lower frustum of the cone = Total volume of cone - Volume of shorter cone = 16 πh 3 - 2 πh 3 cm 3 = 16 πh - 2 πh 3 cm 3 = 14 πh 3 cm 3 ∴ Ratio of two parts of the cone = 2 πh 3 : 14 πh 3 = 2 : 14 = 1 : 7 The ratio of volumes of two parts of the cone is 1 : 7 .

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in figure. If the height of the cylinder is 10 c m , and its base is of radius 3.5 c m . Find the total surface area of the article.

From the given diagram it can be clearly observed that radius of hemisphere is same as radius of cylinder. Given radius of cylindrical part = radius of hemispherical part = r = 3 . 5 cm . Also, height of the cylinder h = 10 cm . The total surface area of the article = ( C S A of cylinder) + ( 2 × CSA of hemisphere) = ( 2 π r h ) + ( 2 × 2 π r 2 ) = ( 2 π × 3 .5 × 10 ) + ( 2 × 2 π × 3 .5 × 3 .5 ) = 70 π + 49 π = 119 × 22 7 = 17 × 22 = 374 cm 2

E M B I B E

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 13 - Surface Areas and Volumes>Exercise>Q 61

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IMPORTANT

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A solid metallic sphere of diameter $28 cm$ is melted and recast into a number of smaller cones, each of diameter $4 \frac{2}{3} cm$ and height $3 cm$ . Find the number of cones so formed.

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Important Questions on Surface Areas and Volumes

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10th Meghalaya Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 13 - Surface Areas and Volumes>Exercise>Q 62

A cube of side

6 cm

is converted into number of cubes of side

2 cm

.The number of cubes is:

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10th Meghalaya Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 13 - Surface Areas and Volumes>Exercise>Q 63

The diameter of a sphere and a right circular cylinder are $12 c m$ . If their volumes are equal then the height of the cylinder will be _____.

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10th Meghalaya Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 13 - Surface Areas and Volumes>Exercise>Q 64

Choose the correct answer:

By melting a right circular cone of height $20 c m$ and diameter of the base $10 c m$ a solid sphere is made. Then the diameter of the sphere will be _____.

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10th Meghalaya Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 13 - Surface Areas and Volumes>Exercise>Q 65

The volume of a solid hemisphere and a solid cylinder are equal. If their diameters are equal, then the ratio of the height and diameter of the cylinder is _____.