Answer :
To solve the problem of finding the radius of the conical water reservoir, we'll use the formula for the volume of a cone. The formula for the volume [tex]\( V \)[/tex] of a cone is given by:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Where:
- [tex]\( V \)[/tex] is the volume of the cone
- [tex]\( \pi \)[/tex] (Pi) is approximately 3.14
- [tex]\( r \)[/tex] is the radius of the base of the cone
- [tex]\( h \)[/tex] is the height of the cone
We know the following from the problem:
- The volume [tex]\( V = 225 \)[/tex] cubic feet
- The height [tex]\( h = 8.5 \)[/tex] feet
We need to determine the radius [tex]\( r \)[/tex]. To do this, we can rearrange the formula to solve for [tex]\( r \)[/tex]:
1. Rearrange the volume formula to solve for [tex]\( r^2 \)[/tex]:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Multiply both sides by 3 to eliminate the fraction:
[tex]\[ 3V = \pi r^2 h \][/tex]
2. Divide both sides by [tex]\( \pi h \)[/tex] to isolate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]
3. Take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{3V}{\pi h}} \][/tex]
Now, substitute the given values (use 3.14 for [tex]\( \pi \)[/tex]):
[tex]\[ r = \sqrt{\frac{3 \times 225}{3.14 \times 8.5}} \][/tex]
When calculated, this gives us a radius of approximately 5.03 feet.
Thus, the correct formula to determine the radius of the water reservoir is:
[tex]\[ r = \sqrt{\frac{3V}{3.14 \times h}} \][/tex]
And the radius, rounded to the nearest hundredth of a foot, is 5.03 feet.
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Where:
- [tex]\( V \)[/tex] is the volume of the cone
- [tex]\( \pi \)[/tex] (Pi) is approximately 3.14
- [tex]\( r \)[/tex] is the radius of the base of the cone
- [tex]\( h \)[/tex] is the height of the cone
We know the following from the problem:
- The volume [tex]\( V = 225 \)[/tex] cubic feet
- The height [tex]\( h = 8.5 \)[/tex] feet
We need to determine the radius [tex]\( r \)[/tex]. To do this, we can rearrange the formula to solve for [tex]\( r \)[/tex]:
1. Rearrange the volume formula to solve for [tex]\( r^2 \)[/tex]:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Multiply both sides by 3 to eliminate the fraction:
[tex]\[ 3V = \pi r^2 h \][/tex]
2. Divide both sides by [tex]\( \pi h \)[/tex] to isolate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]
3. Take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{3V}{\pi h}} \][/tex]
Now, substitute the given values (use 3.14 for [tex]\( \pi \)[/tex]):
[tex]\[ r = \sqrt{\frac{3 \times 225}{3.14 \times 8.5}} \][/tex]
When calculated, this gives us a radius of approximately 5.03 feet.
Thus, the correct formula to determine the radius of the water reservoir is:
[tex]\[ r = \sqrt{\frac{3V}{3.14 \times h}} \][/tex]
And the radius, rounded to the nearest hundredth of a foot, is 5.03 feet.
