It takes 
5
0
 ﻿hours to drain a full cylindrical water storage tank that is 
2
7
 ﻿feet tall. If water is
pumped at a rate of 
3
5
 ﻿gpm 
,
 ﻿then what is the measurement of the tank
'
s radius?

We are tasked with finding the radius of a cylindrical water storage tank based on the given information:

- Time to drain the tank: 50 hours
- Pumping rate: 35 gallons per minute (gpm)
- Height of the tank: 27 feet

### Step 1: Calculate the total volume of water in gallons
We are given that the water is being pumped at a rate of 35 gallons per minute for 50 hours. To find the total amount of water in gallons, we first convert 50 hours into minutes:

\[
50 \text{ hours} = 50 \times 60 = 3000 \text{ minutes}
\]

Now, we calculate the total volume of water in the tank:

\[
\text{Total volume} = 35 \, \text{gallons per minute} \times 3000 \, \text{minutes} = 105,000 \, \text{gallons}
\]

### Step 2: Convert gallons to cubic feet
Since the volume of the tank is given in gallons and we need to work with feet (the height is in feet), we must convert gallons to cubic feet. The conversion factor is:

\[
1 \, \text{gallon} = 0.133681 \, \text{cubic feet}
\]

Therefore, the total volume in cubic feet is:

\[
\text{Total volume in cubic feet} = 105,000 \times 0.133681 = 14,036.505 \, \text{cubic feet}
\]

### Step 3: Use the formula for the volume of a cylinder
The formula for the volume \(V\) of a cylinder is:

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

Where:
- \(r\) is the radius of the tank (which we need to find),
- \(h = 27\) feet is the height of the tank,
- \(V = 14,036.505\) cubic feet is the volume of the tank.

Now, solve for the radius \(r\):

\[
14,036.505 = \pi r^2 \times 27
\]

\[
r^2 = \frac{14,036.505}{\pi \times 27}
\]

\[
r^2 = \frac{14,036.505}{84.823}
\]

\[
r^2 \approx 165.47
\]

\[
r \approx \sqrt{165.47} \approx 12.86 \, \text{feet}
\]

### Final Answer:
The radius of the cylindrical tank is approximately **12.86 feet**.

Would you like further details on any part of the solution or have any other questions?

Here are 5 questions to consider for further exploration:
1. How would the radius change if the pumping rate was reduced to 30 gpm?
2. What is the total surface area of the cylindrical tank?
3. How much time would it take to fill the tank if the pumping rate was doubled?
4. What is the relationship between the radius and the height if the volume is fixed?
5. How does the draining time change if the height of the tank increases by 5 feet?

**Tip:** Always check the units in a problem involving volume to ensure accurate conversions when switching between gallons and cubic feet.

It takes 50 hours to drain a full cylindrical water storage tank that is 27 feet tall. If water is pumped at a rate of 35 gpm, then what is the measurement of the tank's radius?

Discover how to calculate the radius of a cylindrical water tank based on the given height of 27 feet and a draining time of 50 hours at a pumping rate of 35 gpm. Step-by-step conversion from gallons to cubic feet and using the volume of a cylinder formula are included.

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