We are given that the volume of the cylinder remains constant at 120 cubic inches. The volume of a cylinder is given by the formula:

$V = \pi r^2 h$

Where:

• $V$ is the volume,
• $r$ is the radius, and
• $h$ is the height.

Step 1: Differentiate with respect to time

Since the volume $V$ is constant, we differentiate both sides of the equation with respect to time $t$. We need to use implicit differentiation because $r$ and $h$ both depend on time.

$\frac{d}{dt} \left( V = \pi r^2 h \right)$

Using the product rule and the chain rule:

$0 = \pi \left( 2r \frac{dr}{dt} h + r^2 \frac{dh}{dt} \right)$

We are given that:

• The volume $V = 120$ cubic inches (constant),
• $\frac{dh}{dt} = 3$ inches per minute (rate of change of height),
• At the instant when $r = 2$ inches.

Step 2: Solve for $\frac{dr}{dt}$

At the moment when $r = 2$ inches, we can substitute the values into the equation:

$0 = \pi \left( 2(2) \frac{dr}{dt} h + (2)^2 \cdot 3 \right)$

Simplify:

$0 = \pi \left( 4 \frac{dr}{dt} h + 12 \right)$

Now, solve for $\frac{dr}{dt}$:

$4 \frac{dr}{dt} h + 12 = 0$

$4 \frac{dr}{dt} h = -12$

$\frac{dr}{dt} = -\frac{12}{4h}$

$\frac{dr}{dt} = -\frac{3}{h}$

Step 3: Find $h$ using the constant volume condition

Since the volume is constant at 120 cubic inches, we can use the volume formula to solve for $h$ when $r = 2$ inches:

$120 = \pi (2)^2 h$

$120 = 4\pi h$

$h = \frac{120}{4\pi} = \frac{30}{\pi}$

Approximating:

$h \approx \frac{30}{3.1416} \approx 9.425 \, \text{inches}$

Step 4: Calculate $\frac{dr}{dt}$

Substitute $h \approx 9.425$ into the equation for $\frac{dr}{dt}$:

$\frac{dr}{dt} = -\frac{3}{9.425} \approx -0.318 \, \text{inches per minute}$

Thus, the rate of change of the radius at the moment when the radius is 2 inches is approximately:

$\boxed{-0.318} \, \text{inches per minute}$

Final Answer

The rate of change of the radius is approximately $-0.318$ inches per minute.

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Would you like more details or have any other questions?

Here are 5 related questions for further exploration:

1. How does the rate of change of the radius depend on the height of the cylinder?
2. If the height were to increase at a faster rate, how would that affect the rate of change of the radius?
3. What would happen if the volume were not constant? How would the equation change?
4. How would you calculate the rate of change of the volume if the radius were changing at a rate of $0.5$ inches per minute?
5. Can you find the time it would take for the radius to change by 1 inch given the current rate of change?

Tip: Always remember to distinguish between rates of change with respect to time. In problems like this, using implicit differentiation helps account for multiple variables that are all changing over time.