Answer :
To find the radius of the conical water reservoir, we can use the formula for the volume of a cone. The formula for the volume [tex]\( V \)[/tex] of a cone 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 the volume [tex]\( V \)[/tex] is 225 cubic feet and the height [tex]\( h \)[/tex] is 8.5 feet.
To solve for the radius [tex]\( r \)[/tex], we can rearrange the formula to:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]
Then, solve for [tex]\( r \)[/tex] by taking the square root:
[tex]\[ r = \sqrt{\frac{3V}{\pi h}} \][/tex]
Now, let's substitute the known values into the formula:
- [tex]\( V = 225 \)[/tex] cubic feet,
- [tex]\( \pi = 3.14 \)[/tex],
- [tex]\( h = 8.5 \)[/tex] feet.
[tex]\[ r = \sqrt{\frac{3 \times 225}{3.14 \times 8.5}} \][/tex]
After doing the calculations:
[tex]\[ r \approx \sqrt{\frac{675}{26.69}} \][/tex]
[tex]\[ r \approx \sqrt{25.29} \][/tex]
[tex]\[ r \approx 5.03 \][/tex]
Therefore, the radius of the water reservoir is approximately 5.03 feet when rounded to the nearest hundredth of a foot. This matches option:
[tex]\[ r = \sqrt{\frac{3 V}{3.14 h}}, r=5.03 \text{ feet} \][/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 the volume [tex]\( V \)[/tex] is 225 cubic feet and the height [tex]\( h \)[/tex] is 8.5 feet.
To solve for the radius [tex]\( r \)[/tex], we can rearrange the formula to:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]
Then, solve for [tex]\( r \)[/tex] by taking the square root:
[tex]\[ r = \sqrt{\frac{3V}{\pi h}} \][/tex]
Now, let's substitute the known values into the formula:
- [tex]\( V = 225 \)[/tex] cubic feet,
- [tex]\( \pi = 3.14 \)[/tex],
- [tex]\( h = 8.5 \)[/tex] feet.
[tex]\[ r = \sqrt{\frac{3 \times 225}{3.14 \times 8.5}} \][/tex]
After doing the calculations:
[tex]\[ r \approx \sqrt{\frac{675}{26.69}} \][/tex]
[tex]\[ r \approx \sqrt{25.29} \][/tex]
[tex]\[ r \approx 5.03 \][/tex]
Therefore, the radius of the water reservoir is approximately 5.03 feet when rounded to the nearest hundredth of a foot. This matches option:
[tex]\[ r = \sqrt{\frac{3 V}{3.14 h}}, r=5.03 \text{ feet} \][/tex]
