Multiple choice

The radius and height of a cylinder are equal. If the radius of the sphere is equal to the height of the cylinder, then the ratio of the rates of increase of the volume of the sphere and the volume of the cylinder is

  1. $4:3$
  2. $3:4$
  3. $4:3 \pi$
  4. $3:4 \pi $
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A Correct answer
Explanation

Sphere volume V_s = (4/3)pi r^3. Cylinder volume V_c = pi r^2 h. Since r=h, V_c = pi r^3. Rates of increase: dVs/dt = 4 pi r^2 (dr/dt) and dVc/dt = 3 pi r^2 (dr/dt). The ratio is 4/3.

AI explanation

Let the cylinder radius be r, making its height r as well, and let the radius of the sphere also be r. The rate of volume increase for the sphere is 4 pi r squared times the rate of change of r, and for the cylinder it is 2 pi r squared times the rate of change of r. Dividing the sphere's rate by the cylinder's rate gives the ratio 4 to 3.

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