A cylindrical tank gets filled at 88 cubic cm per hour. The level of water in the tank rises at the rate of 7 cm per hour. Find the radius (in cm) of the tank.

Given the problem, we know the following information:

- The rate at which the tank is being filled: \( \frac{dV}{dt} = 88 \) cubic cm per hour.
- The rate at which the water level rises: \( \frac{dh}{dt} = 7 \) cm per hour.

We need to find the radius of the tank. Let's denote the radius by \( r \) cm and the height of the water level by \( h \) cm.

### Step 1: Formula for the volume of a cylinder
The volume \( V \) of a cylinder is given by the formula:

\[
V = \pi r^2 h
\]

### Step 2: Differentiate with respect to time
To relate the rate of volume increase to the rate of height increase, differentiate both sides of the equation with respect to time \( t \):

\[
\frac{dV}{dt} = \pi r^2 \frac{dh}{dt}
\]

### Step 3: Substitute the known values
Now, substitute the given rates:

\[
88 = \pi r^2 \times 7
\]

### Step 4: Solve for \( r \)
Simplify the equation to solve for \( r \):

\[
r^2 = \frac{88}{7\pi}
\]

\[
r^2 = \frac{88}{7 \times 3.1416} \approx 4
\]

\[
r = \sqrt{4} = 2
\]

### Final Answer:
The radius of the tank is \( 2 \) cm.

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

Here are 5 related questions to deepen your understanding:
1. How would the formula change if the tank were a cone instead of a cylinder?
2. What happens to the radius if the rate of water filling doubles?
3. How would you calculate the height of the water after 5 hours?
4. If the radius is doubled, how does it affect the volume of the tank?
5. How would you calculate the volume if the height of the tank were fixed?

**Tip:** Always make sure units are consistent when dealing with rates of change in geometry problems.

This math problem helps you find the radius of a cylindrical tank when the tank is filled at 88 cubic cm per hour and the water level rises at 7 cm per hour. Understand how to use the cylinder volume formula and rate of change to solve it. Suitable for students in Grades 9-12.

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