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Calculate and compare the surface area of sphere A , which has a radius of 2 in., and sphere B , which has a radius of 10 in. The formula for the surface area of a sphere is 4πr^2 .

A. Sphere A has a surface area of 8π in.^2 and sphere B has a surface area of 40π in.^2 , meaning sphere B ’s surface area is 5 times as large as sphere A ’s.

B. Sphere A has a surface area of 4π in.^2 and sphere B has a surface area of 100π in.^2 , meaning sphere B ’s surface area is 25 times as large as sphere A ’s.

C. Sphere A has a surface area of 16π in.^2 and sphere B has a surface area of 400π in.^2 , meaning sphere B ’s surface area is 25 times as large as sphere A ’s.

D. Sphere A has a surface area of 2π in.^2 and sphere B has a surface area of 10π in.^2 , meaning sphere B ’s surface area is 4 times as large as sphere A ’s.

1 answer

To calculate the surface area of the spheres using the formula \( S = 4\pi r^2 \):

For Sphere A:

  • Radius \( r = 2 \) inches
  • Surface area \( S_A = 4\pi (2^2) = 4\pi (4) = 16\pi \) square inches

For Sphere B:

  • Radius \( r = 10 \) inches
  • Surface area \( S_B = 4\pi (10^2) = 4\pi (100) = 400\pi \) square inches

Now, let's compare the surface areas:

  1. The surface area of Sphere A is \( 16\pi \) in².
  2. The surface area of Sphere B is \( 400\pi \) in².
  3. To find how many times larger Sphere B's surface area is than Sphere A's: \[ \text{Ratio} = \frac{S_B}{S_A} = \frac{400\pi}{16\pi} = \frac{400}{16} = 25 \]

Thus, Sphere B’s surface area is 25 times as large as Sphere A’s.

In conclusion, the correct choice is:

C. Sphere A has a surface area of 16π in² and sphere B has a surface area of 400π in², meaning sphere B’s surface area is 25 times as large as sphere A's.