Answer :
To determine the radius of a conical water reservoir, we start with the formula for the volume of a cone, which is:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
In this formula:
- [tex]\( V \)[/tex] is the volume of the cone, which is given as 225 cubic feet.
- [tex]\( \pi \)[/tex] is a constant, approximately 3.14.
- [tex]\( r \)[/tex] is the radius of the base of the cone, which we are trying to find.
- [tex]\( h \)[/tex] is the height of the cone, given as 8.5 feet.
We need to rearrange this formula to solve for the radius [tex]\( r \)[/tex].
1. Multiply both sides of the equation by 3 to eliminate the fraction:
[tex]\[ 3V = \pi r^2 h \][/tex]
2. Next, we need to isolate [tex]\( r^2 \)[/tex]. Divide both sides by [tex]\( \pi h \)[/tex]:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]
3. To solve for [tex]\( r \)[/tex], take the square root of both sides:
[tex]\[ r = \sqrt{\frac{3V}{\pi h}} \][/tex]
Plug in the known values:
- [tex]\( V = 225 \)[/tex]
- [tex]\( h = 8.5 \)[/tex]
- Approximate [tex]\( \pi \)[/tex] as 3.14 for simplicity.
Now calculate the radius:
[tex]\[ r = \sqrt{\frac{3 \times 225}{3.14 \times 8.5}} \][/tex]
After calculating the expression, you find the radius [tex]\( r \)[/tex] to be approximately 5.03 feet when rounded to the nearest hundredth.
So, the formula [tex]\( r = \sqrt{\frac{3V}{3.14 \times h}} \)[/tex] is used, and the radius of the water reservoir is 5.03 feet, rounded to the nearest hundredth.
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
In this formula:
- [tex]\( V \)[/tex] is the volume of the cone, which is given as 225 cubic feet.
- [tex]\( \pi \)[/tex] is a constant, approximately 3.14.
- [tex]\( r \)[/tex] is the radius of the base of the cone, which we are trying to find.
- [tex]\( h \)[/tex] is the height of the cone, given as 8.5 feet.
We need to rearrange this formula to solve for the radius [tex]\( r \)[/tex].
1. Multiply both sides of the equation by 3 to eliminate the fraction:
[tex]\[ 3V = \pi r^2 h \][/tex]
2. Next, we need to isolate [tex]\( r^2 \)[/tex]. Divide both sides by [tex]\( \pi h \)[/tex]:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]
3. To solve for [tex]\( r \)[/tex], take the square root of both sides:
[tex]\[ r = \sqrt{\frac{3V}{\pi h}} \][/tex]
Plug in the known values:
- [tex]\( V = 225 \)[/tex]
- [tex]\( h = 8.5 \)[/tex]
- Approximate [tex]\( \pi \)[/tex] as 3.14 for simplicity.
Now calculate the radius:
[tex]\[ r = \sqrt{\frac{3 \times 225}{3.14 \times 8.5}} \][/tex]
After calculating the expression, you find the radius [tex]\( r \)[/tex] to be approximately 5.03 feet when rounded to the nearest hundredth.
So, the formula [tex]\( r = \sqrt{\frac{3V}{3.14 \times h}} \)[/tex] is used, and the radius of the water reservoir is 5.03 feet, rounded to the nearest hundredth.
