If the surface area of a cube with side x cm is increasing at a rate of 64 cm^2/s, find the rate of change of x when the surface area is 384 cm^2.

To solve this problem, we need to relate the rate at which the surface area of the cube is changing to the rate at which the side length \( x \) of the cube is changing.

The surface area \( S \) of a cube with side length \( x \) is given by:
\[ S = 6x^2 \]

We are given that the surface area is increasing at a rate of 64 cm²/s. This means:
\[ \frac{dS}{dt} = 64 \, \text{cm}^2/\text{s} \]

We need to find the rate at which the side length \( x \) is changing when the surface area \( S \) is 384 cm². Let's denote this rate by \( \frac{dx}{dt} \).

First, we differentiate the surface area \( S \) with respect to time \( t \):
\[ \frac{dS}{dt} = \frac{d}{dt} (6x^2) \]

Using the chain rule:
\[ \frac{dS}{dt} = 12x \frac{dx}{dt} \]

We can now plug in the given values:
\[ 64 = 12x \frac{dx}{dt} \]

Next, we need to determine the value of \( x \) when the surface area is 384 cm²:
\[ 6x^2 = 384 \]
\[ x^2 = \frac{384}{6} \]
\[ x^2 = 64 \]
\[ x = 8 \, \text{cm} \]

Now, we can solve for \( \frac{dx}{dt} \):
\[ 64 = 12 \cdot 8 \cdot \frac{dx}{dt} \]
\[ 64 = 96 \cdot \frac{dx}{dt} \]
\[ \frac{dx}{dt} = \frac{64}{96} \]
\[ \frac{dx}{dt} = \frac{2}{3} \, \text{cm/s} \]

So, the rate of change of \( x \) when the surface area is 384 cm² is \( \frac{2}{3} \, \text{cm/s} \).

Would you like more details or have any questions? Here are some related questions to explore further:

1. How does the rate of change of the side length vary as the surface area increases?
2. What is the volume of the cube when the surface area is 384 cm²?
3. How long will it take for the side length to increase by 1 cm at the given rate?
4. What is the relationship between the rate of change of surface area and the rate of change of the volume of the cube?
5. How does the surface area change if the side length is increasing at a constant rate?
6. What would happen to the rate of change of the side length if the rate of change of the surface area were doubled?
7. Can the side length be decreasing if the surface area is increasing? Why or why not?
8. How would you set up and solve the related rates problem if the surface area were given in square inches per second?

**Tip:** When dealing with related rates problems involving geometry, always express the quantities in terms of the variable you're interested in, then use differentiation to relate the rates.

Learn how to solve a related rates problem involving the surface area of a cube and the rate of change of its side length. Find the rate of change of x when the cube's surface area is 384 cm². Detailed step-by-step solution provided.

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