Question

A mathematics teacher uses certain amount of terracotta clay to form different shaped solids. First, she turned it into a sphere of radius 7 cm and then she made a right circular cone with base radius 14 cm. Find the height of the cone so formed. If the same clay is turned to make a right circular cylinder of height `7/3` cm, then find the radius of the cylinder so formed. Also, compare the total surface areas of sphere and cylinder so formed.

Answer 1

First, a sphere of radius (r) 7 cm is formed.

Volume of sphere = `4/3 πr^3`

= `4/3 xx 22/7 xx 7^3`

= `4/3 xx 22 xx 7^2`

= `4312/3 cm^3`

Next, a right circular cone with radius (r₁) 14 cm is formed. Let height of cone be h cm.

Since same amount of clay is used to make cone and sphere.

∴ Volume of cone = Volume of sphere

⇒ `1/3 πr_1^2h = 4312/3`

⇒ `πr_1^2h = 4312`

⇒ `22/7 xx 14^2 xx h = 4312`

⇒ 22 × 2 × 14 × h = 4312

⇒ `h = 4312/(22 xx 2 xx 14)`

⇒ `h = 4312/616`

⇒ h = 7 cm

Given,

The same clay is used to make a right circular cylinder of height (h₁) `7/3` cm. Let its radius be r₂.

Since same amount of clay is used to make cylinder and sphere.

∴ Volume of cylinder = Volume of sphere

⇒ `πr_2^2h_1 = 4312/3`

⇒ `22/7 xx r_2^2 xx 7/3 = 4312/3`

⇒ `22 xx r_2^2 = 4312`

⇒ `r_2^2 = 4312/22`

⇒ `r_2^2 = 196`

⇒ `r_2 = sqrt(196)`

⇒ r = 14 cm

Total surface area of sphere = 4πr²

Total surface area of cylinder = 2πr₂(r₂ + h₁)

⇒ `("TSA sphere")/("TSA cylinder") = (4πr^2)/(2πr_2(r_2 + h_1))`

= `(2r^2)/(r_2(r_2 + h_1))`

= `(2 xx 7^2)/(14(14 + 7/3))`

= `98/(14 xx 49/3)`

= `(98 xx 3)/(14 xx 49)`

= `3/7`

Hence, height of cone = 7 cm, radius of cylinder = 14 cm and Total surface area of sphere : Total surface area of cylinder = 3 : 7.