Answer :
To find the radius of the conical water reservoir, we'll use the formula for the volume of a cone. The formula is:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Where:
- [tex]\( V \)[/tex] is the volume of the cone,
- [tex]\( r \)[/tex] is the radius of the base of the cone,
- [tex]\( h \)[/tex] is the height of the cone,
- [tex]\( \pi \)[/tex] is approximately 3.14.
We know from the problem that:
- The volume of the reservoir [tex]\( V \)[/tex] is 225 cubic feet,
- The height [tex]\( h \)[/tex] is 8.5 feet.
We need to solve for [tex]\( r \)[/tex], the radius of the base:
1. Rearrange the formula to solve for [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]
2. Substitute the known values into the formula:
[tex]\[ r^2 = \frac{3 \times 225}{3.14 \times 8.5} \][/tex]
3. Calculate the value on the right side:
[tex]\[ r^2 = \frac{675}{26.69} \][/tex]
4. Simplify the division:
[tex]\[ r^2 \approx 25.30 \][/tex]
5. Take the square root to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{25.30} \][/tex]
6. Find the square root:
[tex]\[ r \approx 5.03 \][/tex]
Therefore, the radius of the water reservoir, rounded to the nearest hundredth of a foot, is approximately 5.03 feet. This matches the formula [tex]\( r = \sqrt{\frac{3V}{3.14h}} \)[/tex], so the correct option is:
[tex]\[ r = \sqrt{\frac{3V}{3.14h}}, \text{ with } r = 5.03 \text{ feet} \][/tex]
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Where:
- [tex]\( V \)[/tex] is the volume of the cone,
- [tex]\( r \)[/tex] is the radius of the base of the cone,
- [tex]\( h \)[/tex] is the height of the cone,
- [tex]\( \pi \)[/tex] is approximately 3.14.
We know from the problem that:
- The volume of the reservoir [tex]\( V \)[/tex] is 225 cubic feet,
- The height [tex]\( h \)[/tex] is 8.5 feet.
We need to solve for [tex]\( r \)[/tex], the radius of the base:
1. Rearrange the formula to solve for [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]
2. Substitute the known values into the formula:
[tex]\[ r^2 = \frac{3 \times 225}{3.14 \times 8.5} \][/tex]
3. Calculate the value on the right side:
[tex]\[ r^2 = \frac{675}{26.69} \][/tex]
4. Simplify the division:
[tex]\[ r^2 \approx 25.30 \][/tex]
5. Take the square root to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{25.30} \][/tex]
6. Find the square root:
[tex]\[ r \approx 5.03 \][/tex]
Therefore, the radius of the water reservoir, rounded to the nearest hundredth of a foot, is approximately 5.03 feet. This matches the formula [tex]\( r = \sqrt{\frac{3V}{3.14h}} \)[/tex], so the correct option is:
[tex]\[ r = \sqrt{\frac{3V}{3.14h}}, \text{ with } r = 5.03 \text{ feet} \][/tex]
