Question

the ratio of the volumes of two spheres is 27:8. find the ratio of their surface areas.

Answer 1

Let r1 and r2 be the radius of the two spheres.
4/3 π r1³ : 4/3 π r2³ = 27 : 8
r1³ : r2³ = 27 : 8
r1 : r2 = ∛ 27 : 8
r1 : r2 = 3 : 2

Ratio of surface area = 4πr1² : 4πr2² 
= r1² : r2² = 3² : 2² = 9 : 4

Answer 2

Given,

The ratio of the volumes of two spheres is 27:8.

To find,

The ratio of their surface areas.

Solution,

We can simply solve this mathematical problem using the following process:

Let us assume that the radius of the first sphere be R1 and the radius of the second sphere be R2.

As per mensuration;

The volume of a sphere = 4/3πr^3

The surface area of a sphere = 4πr^2

Now,

The ratio of the volumes of two spheres = 27:8

=> (volume of the first sphere)/(volume of the second sphere) = 27/8

=> {4/3π(R1)^3}/{4/3π(R2)^3} = 27/8

=> (R1)^3/(R2)^3 = (R1/R2)^3 = (3/2)^3

=> R1/R2 = 3/2

{Equation-1}

Now, the ratio of their surface areas

= {4π(R1)^2}/{4π(R2)^2}

= (R1)^2/(R2)^2

= (R1/R2)^2

= (3/2)^2 {using equation-1}

= 9/4

Hence, the ratio of the surface areas of both the spheres is equal to 9/4.