`T=(R^(2))/(r^(2))sqrt((2h)/(g))`
Suppose an open cylindrical tank has a round drain with radius t in the bottom of the tank. When the tank is filled with water to a depth of h centimeters, the time it takes for all the water to drain from the tank is given by the formula above, where R is the radius of the tank (in centimeters) and `g=980"cm/"s^(2)` is the acceleration due to gravity. Suppose such a tank has a radius of 2 meters and is filled to a depth of 4 meters. About how many minutes does it take to empty the tank if the drain has a radius of 5 centimeters? (1 meter=100 centimeter).

Explanation

To solve the problem, we will use the formula provided and substitute the given values step by step.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Radius of the tank (R) = 2 meters = 200 centimeters (since 1 meter = 100 centimeters)
- Radius of the drain (r) = 5 centimeters
- Depth of the water (h) = 4 meters = 400 centimeters
- Acceleration due to gravity (g) = 980 cm/s²

2. **Write the Formula**:
The time \( T \) to drain the tank is given by the formula:
\[ T = \frac{R^2}{r^2} \sqrt{\frac{2h}{g}} \]

3. **Substitute the Values**:
Substitute \( R = 200 \) cm, \( r = 5 \) cm, \( h = 400 \) cm, and \( g = 980 \) cm/s² into the formula:
\[ T = \frac{200^2}{5^2} \sqrt{\frac{2 \times 400}{980}} \]

4. **Calculate \( R^2 \) and \( r^2 \)**:
- \( R^2 = 200^2 = 40000 \)
- \( r^2 = 5^2 = 25 \)

5. **Calculate the Fraction**:
\[ \frac{R^2}{r^2} = \frac{40000}{25} = 1600 \]

6. **Calculate the Square Root**:
First, calculate the numerator of the square root:
\[ 2h = 2 \times 400 = 800 \]
Now calculate the square root:
\[ \sqrt{\frac{800}{980}} = \sqrt{\frac{800}{980}} = \sqrt{\frac{80}{98}} = \sqrt{\frac{8}{9.8}} \approx \sqrt{0.8163} \approx 0.9035 \]

7. **Combine the Results**:
Now substitute back into the equation for \( T \):
\[ T = 1600 \times 0.9035 \approx 1445.6 \text{ seconds} \]

8. **Convert Seconds to Minutes**:
To convert seconds to minutes, divide by 60:
\[ T \approx \frac{1445.6}{60} \approx 24.09 \text{ minutes} \]
Rounding down, we get approximately 24 minutes.

### Final Answer:
It takes about **24 minutes** to empty the tank.