Multiple choice

The sum of radii spheres A and B is 14 cm, the radius of A being larger than that of B. The difference between their surface areas is (112 \pi). What is the ratio of the volumes of A and B?

  1. $64 : 27$
  2. $8 : 1$
  3. $125 : 64$
  4. $8 : 27$
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A Correct answer
Explanation

Let r₁, r₂ be radii with r₁ > r₂. r₁ + r₂ = 14. Surface area difference: 4π(r₁² - r₂²) = 112π → 4π(r₁ - r₂)(r₁ + r₂) = 112π → 56(r₁ - r₂) = 112 → r₁ - r₂ = 2. Solving: r₁ = 8, r₂ = 6. Volume ratio = r₁³:r₂³ = 512:216 = 64:27.

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