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The surface area of a sphere of radius 5 cm is five times the area of the curved surface of a cone of radius 4 cm. Find the height and the volume of the cone (taking π = 22/7).

Solution:

Given, the radius of cone = 4 cm

The radius of sphere = 5 cm

The surface area of the sphere is five times the curved surface area of the cone.

We have to find the height and volume of the cone.

Surface area of the sphere = 4πr²

Where, r is the radius of the sphere

Given, r = 5 cm

Surface area of sphere = 4π(5)²

= 4(25)π

= 100π cm²

Curved surface area of cone = πrl

Where, r is the radius of the cone

l is the slant height of the cone

Given, r = 4 cm

Curved surface area = π(4)l

= 4πl cm²

Given, surface area of sphere = 5 (curved surface area of cone)

100π = 5(4πl)

100 = 5(4l)

20l = 100

l = 100/20

l = 5 cm

We know, l² = r² + h²

(5)² = (4)² + h²

25 = 16 + h²

h² = 25 - 16

h² = 9

Taking square root,

h = 3 cm

Volume of the cone = 1/3 πr²h

= 1/3 (22/7)(4)²(3)

= (22/7)(16)

= 352/7

= 50.29 cm²

Therefore, the volume of cone is 50.29 cm²

✦ Try This: The surface area of a sphere of radius 3 cm is three times the area of the curved surface of a cone of radius 5 cm. Find the height and the volume of the cone.

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 13

NCERT Exemplar Class 9 Maths Exercise 13.3 Sample Problem 1

The surface area of a sphere of radius 5 cm is five times the area of the curved surface of a cone of radius 4 cm. Find the height and the volume of the cone (taking π = 22/7).

Summary:

The surface area of a sphere of radius 5 cm is five times the area of the curved surface of a cone of radius 4 cm. The height and the volume of the cone is 3 cm and 50.29 cm²

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