Answer is: option3

15 in³/sec

Solution:

We are given a cube where the edge length is increasing at a rate of:

$$\frac{dx}{dt} = 0.2 \text{ in/sec}$$

We need to determine the rate of change of volume, $$\frac{dV}{dt}$$, when the total surface area is 150 square inches.

The surface area of a cube is given by:

$$S = 6x^2$$

Setting $$S = 150$$:

$$6x^2 = 150$$

$$x^2 = 25$$

$$x = 5 \text{ inches}$$

The volume of a cube is:

$$V = x^3$$

Differentiate both sides with respect to $$t$$:

$$\frac{dV}{dt} = 3x^2 \frac{dx}{dt}$$

Substituting $$x = 5$$ and $$\frac{dx}{dt} = 0.2$$:

$$\frac{dV}{dt} = 3(5^2)(0.2)$$

$$= 3(25)(0.2)$$

$$= 15 \text{ in}^3/\text{sec}$$

The correct answer is 15 in³/sec.