PSAT Math Multiple-Choice Question 720: Answer and Explanation

Question: 720

The formula for the surface area of a sphere is A = 4πr². If the volume of sphere A is 8 times the volume of sphere B , what is the ratio of the surface area of sphere A to that of sphere B ?

• A. 1:2
• B. 2:1
• C. 4:1
• D. 8:1

Correct Answer: C

Explanation:

(C) We're given that V_A = 8V_B . From the formula for the volume of a sphere , it follows that . The term cancels out on both sides, leaving us with Taking the cube root of both sides gives r_A = 2r_B . Thus, the radius of sphere A is twice the radius of sphere B. We ultimately want to find the ratio of the surface areas, so ­consider the surface areas of the two spheres:

and

To find the ratio of the surface area of sphere A to the surface area of sphere B, we divide the surface area of sphere A by the surface area of sphere B:

We can cancel a 4π and an out of both the numerator and the denominator, which leaves us with .

Thus, the ratio of the surface area of sphere A to sphere B is or 4:1, which is choice (C).