The volume of a cube is increasing at the rate of 6cm3/s. How fast is the surface 2 area of cube increasing, when the length of an edge is cm?

To solve this, we need to use related rates involving the volume and surface area of a cube. Let \( x \) be the length of an edge of the cube (in cm), \( V \) the volume, and \( A \) the surface area.

### Step 1: Relations for Volume and Surface Area

1. The volume of a cube is given by:
   \[
   V = x^3
   \]
2. The surface area of a cube is given by:
   \[
   A = 6x^2
   \]

### Step 2: Differentiating Volume and Surface Area with Respect to Time

1. Differentiate the volume equation \( V = x^3 \) with respect to time \( t \):
   \[
   \frac{dV}{dt} = 3x^2 \frac{dx}{dt}
   \]
   We are given that \( \frac{dV}{dt} = 6 \, \text{cm}^3/\text{s} \).

2. Differentiate the surface area equation \( A = 6x^2 \) with respect to time \( t \):
   \[
   \frac{dA}{dt} = 12x \frac{dx}{dt}
   \]

### Step 3: Solve for \( \frac{dx}{dt} \)

Using the equation from the volume:
\[
6 = 3x^2 \frac{dx}{dt}
\]
Solve for \( \frac{dx}{dt} \):
\[
\frac{dx}{dt} = \frac{6}{3x^2} = \frac{2}{x^2}
\]

### Step 4: Substitute \( \frac{dx}{dt} \) into the Surface Area Equation

Now, substitute \( \frac{dx}{dt} = \frac{2}{x^2} \) into the equation for \( \frac{dA}{dt} \):
\[
\frac{dA}{dt} = 12x \times \frac{2}{x^2} = \frac{24}{x}
\]

So, the rate at which the surface area is increasing is:
\[
\frac{dA}{dt} = \frac{24}{x} \, \text{cm}^2/\text{s}
\]

### Final Answer:
The surface area is increasing at the rate of \( \frac{24}{x} \, \text{cm}^2/\text{s} \) when the edge length is \( x \) cm.

Would you like to go over any step in more detail, or explore some variations of this problem?

Here are 5 related questions:
1. What happens if the rate of change of the volume was different?
2. How would the problem change if the shape were a sphere?
3. What is the rate of change of the diagonal of the cube?
4. How fast is the edge length increasing when the surface area is increasing at 10 cm²/s?
5. How does the surface area change when the edge is doubled?

**Tip:** When solving related rates problems, make sure to differentiate each variable with respect to time carefully!

The volume of a cube is increasing at the rate of 6 cm³/s. How fast is the surface area of the cube increasing, when the length of an edge is x cm?

Solve a related rates problem involving the volume and surface area of a cube. Learn how to calculate how fast the surface area of a cube increases when the volume changes at 6 cm³/s. Suitable for Grades 11-12 students studying calculus and geometry.

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