Ratio of the volume and the surface area between two spheres

by Rabbani

Question

The radius of sphere A is twice that of a sphere B. Find the ratio among their surface areas and the ratio among their volumes.

Answer

Consider the spheres below. These two spheres have the radius of 2r and r respectively.
 
The ratio of  the surface area of sphere A to sphere B

Let the surface area of sphere A be S_A and the surface area of sphere B be S_B. Now we know that the formula for the surface area of sphere S is:  Therefore, to find S_A, we just need to substitute r with 2r and simplify the equation. This is shown below: Similarly, to find S_B, since the radius is r, we have:
 Hence, the ratio of S_A to S_B will be: 
The ratio of  the volume of sphere A to sphere B

Let the volume of sphere A be V_A and the volume of sphere B be V_B. Now, we know that the formula for the volume of a sphere is: To find V_A, we just substitute r with 2r and simplify the equation. This is shown below: Next, to find V_B, since the radius of sphere B is r, we have: 
Hence, the ratio of V_A to V_B is: