Question

The side of a cube is equal to the diameter of a sphere. If the side and radius increase at the same rate then the ratio of the increase of their surface area is

• A. \( 3 : \pi \)
• B. \( \pi : 6 \)
• C. \( 2\pi : 3 \)
• D. \( 3 : 2\pi \)

Hint:

When solving problems involving rates of change of geometric quantities, remember to differentiate the relevant formulas with respect to time and use the given relationships to simplify the expressions.

Answer 1

The Correct Option is A

Let the side of the cube be \( s \), and the radius of the sphere be \( r \). Given that the side of the cube is equal to the diameter of the sphere, we have: \[ s = 2r \] Now, we are asked to find the ratio of the increase in their surface areas. 1. Surface area of the cube: The surface area \( A_{\text{cube}} \) of a cube with side length \( s \) is given by: \[ A_{\text{cube}} = 6s^2 \] 2. Surface area of the sphere: The surface area \( A_{\text{sphere}} \) of a sphere with radius \( r \) is given by: \[ A_{\text{sphere}} = 4\pi r^2 \] Next, we calculate the rate of increase in surface area. Let \( ds/dt \) represent the rate of change of the side of the cube, and \( dr/dt \) represent the rate of change of the radius of the sphere. Since the side and the radius are increasing at the same rate, we have: \[ \frac{ds}{dt} = \frac{dr}{dt} \] To find the rate of increase of the surface areas, we differentiate both surface area formulas with respect to time. 1. Rate of change of surface area of the cube: \[ \frac{d}{dt}(A_{\text{cube}}) = \frac{d}{dt}(6s^2) = 12s \frac{ds}{dt} \] 2. Rate of change of surface area of the sphere: \[ \frac{d}{dt}(A_{\text{sphere}}) = \frac{d}{dt}(4\pi r^2) = 8\pi r \frac{dr}{dt} \] Now, we compute the ratio of the rate of increase of the surface areas: \[ \frac{\frac{d}{dt}(A_{\text{cube}})}{\frac{d}{dt}(A_{\text{sphere}})} = \frac{12s \frac{ds}{dt}}{8\pi r \frac{dr}{dt}} = \frac{12s}{8\pi r} \] Since \( s = 2r \), we substitute into the equation: \[ \frac{12(2r)}{8\pi r} = \frac{24r}{8\pi r} = \frac{3}{\pi} \] Thus, the ratio of the increase in their surface areas is \( 3 : \pi \). Hence, the correct answer is (A).