Question

If the volumes of two spheres are in the ratio 125 : 216,
then the respective ratio of their surface areas is​

Answer 1

Answer:

ratio of surface area is 25/36

Answer 2

Solution

Given :-

• the volumes of two spheres are in the ratio = 125 : 216

Find :-

• Ratio of thier surface area.

Explanation

Let,

• Radius of first sphere = r
• Radius of second sphere = r'
• Volume of first sphere = V
• Volume of second sphere = V'
• Surface area of first sphere = SA
• Surface area of second sphere = SA'

Using Formula

$\boxed{\underline{\tt{\red{\:Volume_{sphere}\:=\:\dfrac{4\pi\:r^3}{3}}}}}$

$\boxed{\underline{\tt{\orange{\: Surface\:Area_{sphere}\:=\:4\pi\:r^2}}}}$

Case 1.

==> Volume of first sphere (V) = 4πr³/3

And,

==> Volume of second sphere (V') = 4πr'³/3

According to question,

==> V/V' = (4πr³/3)/(4πr'³/3)

==> 125 /216 = (r/r')³

==> (5/6)³ = (r/r')³

We know,

• if a^m = b^m , then always be a = b .

Use above property,

==> r/r' = 5)6 _________________(1)

Case 2.

==> Surface area of first sphere (SA) = 4πr²

And,

==> Surface area of second sphere (SA') = 4πr'²

Take ratio of both surface area.

==> SA/SA' = 4πr²/4πr'²

==> SA/SA' = (r/r')²

Keep value by equ(1)

==> SA/SA' = (5/6)²

==> SA/SA' = 25/36

Or,

==> SA : SA' = 25 : 36

Hence

• ratio of their surface areas will be = 25 : 36

__________________