To determine how many hours it takes to fill a cylindrical pool with a given hose flow rate, follow these steps:
1. Calculate the volume of the pool in cubic feet:
- The pool is a cylinder, so use the formula for the volume of a cylinder:
[tex]\[
V = \pi r^2 h
\][/tex]
where:
- [tex]\(r\)[/tex] is the radius of the base of the cylinder, which is 20 feet.
- [tex]\(h\)[/tex] is the height of the cylinder, which is 4.1 feet.
- Plugging in these values:
[tex]\[
V = \pi \times (20)^2 \times 4.1 = 5152.21 \text{ cubic feet}
\][/tex]
2. Convert the volume from cubic feet to gallons:
- Use the conversion factor where 1 cubic foot is approximately 7.48052 gallons.
[tex]\[
\text{Volume in gallons} = 5152.21 \text{ cubic feet} \times 7.48052 = 38541.22 \text{ gallons}
\][/tex]
3. Determine the time to fill the pool in minutes:
- The hose has a flow rate of 13 gallons per minute (gpm).
[tex]\[
\text{Time in minutes} = \frac{\text{Volume in gallons}}{\text{Flow rate in gpm}} = \frac{38541.22}{13} = 2964.71 \text{ minutes}
\][/tex]
4. Convert the time from minutes to hours:
- Since there are 60 minutes in an hour:
[tex]\[
\text{Time in hours} = \frac{\text{Time in minutes}}{60} = \frac{2964.71}{60} = 49.41 \text{ hours}
\][/tex]
5. Round the result to the tenths place:
- Rounding 49.41 to the nearest tenth results in:
[tex]\[
\text{Time in hours (rounded)} = 49.4 \text{ hours}
\][/tex]
Hence, it takes approximately 49.4 hours to fill the cylindrical pool.
