Question

A cube is expanding in such a way that its edge is increasing at a rate of 2 inches per second. If its edge is 5 inches long, then the rate of change of its volume is

• A. 150 in³/sec
• B. 75 in³/sec
• C. 50 in³/sec
• D. 30 in³/sec
• E. 45 in³/sec

Answer 1

The Correct Option is A

We are given that a cube is expanding such that its edge is increasing at a rate of 2 inches per second, and the length of the edge is 5 inches. We are asked to find the rate of change of its volume.

The volume \( V \) of a cube is given by the formula: 

\( V = s^3 \),

where \( s \) is the length of the edge of the cube.

We are asked to find the rate of change of the volume, \( \frac{dV}{dt} \), when the edge is 5 inches long, and the rate of change of the edge is \( \frac{ds}{dt} = 2 \) inches per second.

To find \( \frac{dV}{dt} \), we differentiate \( V = s^3 \) with respect to \( t \) using the chain rule:

\( \frac{dV}{dt} = 3s^2 \frac{ds}{dt} \).

Substitute \( s = 5 \) and \( \frac{ds}{dt} = 2 \) into this equation:

\( \frac{dV}{dt} = 3(5)^2 \cdot 2 = 3 \cdot 25 \cdot 2 = 150 \) cubic inches per second.

The correct answer is 150 in³/sec.