The volume of a cube is increasing at the rate of 16 cm3/s. At what rate is its total surface area increasing when the length of an edge is 6 cm?

To find the rate at which the total surface area of the cube is increasing, we can use calculus, specifically related rates. Let's break down the problem:

1. **Volume of a cube**:  
   The volume \( V \) of a cube with edge length \( s \) is given by:
   \[
   V = s^3
   \]

2. **Surface area of a cube**:  
   The total surface area \( A \) of a cube with edge length \( s \) is given by:
   \[
   A = 6s^2
   \]

3. **Given information**:
   - The rate at which the volume is increasing: \(\frac{dV}{dt} = 16 \, \text{cm}^3/\text{s}\).
   - The edge length \( s = 6 \, \text{cm} \).

4. **Objective**:  
   Find the rate at which the surface area is increasing: \(\frac{dA}{dt}\).

### Step-by-Step Solution:

#### Step 1: Differentiate the volume equation with respect to time \( t \).

The derivative of \( V = s^3 \) with respect to time is:
\[
\frac{dV}{dt} = 3s^2 \frac{ds}{dt}
\]

#### Step 2: Solve for \(\frac{ds}{dt}\).

Given that \(\frac{dV}{dt} = 16 \, \text{cm}^3/\text{s}\):
\[
16 = 3(6)^2 \frac{ds}{dt}
\]
\[
16 = 108 \frac{ds}{dt}
\]
\[
\frac{ds}{dt} = \frac{16}{108} = \frac{4}{27} \, \text{cm/s}
\]

#### Step 3: Differentiate the surface area equation with respect to time \( t \).

The derivative of \( A = 6s^2 \) with respect to time is:
\[
\frac{dA}{dt} = 12s \frac{ds}{dt}
\]

#### Step 4: Substitute \( s = 6 \, \text{cm} \) and \(\frac{ds}{dt} = \frac{4}{27} \, \text{cm/s}\) into the surface area derivative.

\[
\frac{dA}{dt} = 12 \times 6 \times \frac{4}{27}
\]
\[
\frac{dA}{dt} = 72 \times \frac{4}{27}
\]
\[
\frac{dA}{dt} = \frac{288}{27}
\]
\[
\frac{dA}{dt} = \frac{32}{3} \, \text{cm}^2/\text{s}
\]

Thus, the rate at which the total surface area of the cube is increasing when the edge length is 6 cm is \(\frac{32}{3} \, \text{cm}^2/\text{s}\).

---

Would you like more details on any step, or do you have any questions?

### Related Questions

1. How do you find the rate of change of the diagonal of a cube when the edge length changes?
2. What is the rate of change of the face diagonal of a cube?
3. How would the solution change if the cube were expanding at a different rate?
4. Can you apply similar methods to find the rate of change of other geometric shapes?
5. How does changing the rate of volume affect the rate of surface area change?
6. How do you solve a related rates problem for a cylinder or a sphere?
7. What happens to the rates of change if the cube's volume is decreasing instead?
8. How can you verify your solution to a related rates problem?

### Tip

When solving related rates problems, always identify what rates are given and what rates need to be found, and remember to express each variable as a function of time if it changes over time.

Discover how to calculate the rate at which the total surface area of a cube increases when the edge length is 6 cm. This detailed solution uses calculus and related rates to derive the rate of change formula. Ideal for advanced high school students and anyone interested in calculus applications.

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