Sure, let's work through the problem step-by-step to determine how many hours it takes to fill the cylindrical pool using the given values.
### Step 1: Find the volume of the cylindrical pool in cubic feet.
The formula to find the volume [tex]\( V \)[/tex] of a cylinder is given by:
[tex]\[ V = \pi \times r^2 \times h \][/tex]
Where:
- [tex]\( r \)[/tex] is the radius of the base,
- [tex]\( h \)[/tex] is the height, and
- [tex]\( \pi \)[/tex] is approximately 3.14159.
Given:
- Height [tex]\( h = 4.1 \)[/tex] feet,
- Radius [tex]\( r = 20 \)[/tex] feet.
Plugging in these values:
[tex]\[ V = \pi \times (20)^2 \times 4.1 \approx 5152.2119518872605 \text{ cubic feet} \][/tex]
### Step 2: Convert the volume from cubic feet to gallons.
1 cubic foot is approximately equal to 7.481 gallons.
[tex]\[ \text{Volume in gallons} = 5152.2119518872605 \text{ cubic feet} \times 7.481 \approx 38543.697612068594 \text{ gallons} \][/tex]
### Step 3: Calculate the time required to fill the pool in minutes.
Using the flow rate of the hose, which is 13 gallons per minute, we can find the time in minutes needed to fill the pool.
[tex]\[ \text{Time in minutes} = \frac{\text{Total volume in gallons}}{\text{Flow rate in gallons per minute}} \][/tex]
[tex]\[ \text{Time in minutes} = \frac{38543.697612068594 \text{ gallons}}{13 \text{ gpm}} \approx 2964.8998163129686 \text{ minutes} \][/tex]
### Step 4: Convert the time from minutes to hours.
There are 60 minutes in an hour.
[tex]\[ \text{Time in hours} = \frac{\text{Time in minutes}}{60} \][/tex]
[tex]\[ \text{Time in hours} = \frac{2964.8998163129686 \text{ minutes}}{60} \approx 49.41499693854948 \text{ hours} \][/tex]
### Step 5: Round to the tenths place.
Rounding 49.41499693854948 to the tenths place, we get:
[tex]\[ \text{Time in hours (rounded)} = 49.4 \][/tex]
Thus, it takes approximately 49.4 hours to fill the pool.
