Answer :
To determine the radius of the conical water reservoir, we need to use the formula for the volume of a cone. The volume [tex]\( V \)[/tex] of a cone can be calculated using the formula:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Where:
- [tex]\( V \)[/tex] is the volume
- [tex]\( r \)[/tex] is the radius of the base
- [tex]\( h \)[/tex] is the height of the cone
- [tex]\( \pi \)[/tex] is approximately 3.14
From the problem, we know:
- The volume [tex]\( V = 225 \)[/tex] cubic feet
- The height [tex]\( h = 8.5 \)[/tex] feet
Our goal is to solve for [tex]\( r \)[/tex], the radius. First, rearrange the formula to solve for [tex]\( r^2 \)[/tex]:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Multiply both sides by 3 to get rid of the fraction:
[tex]\[ 3V = \pi r^2 h \][/tex]
Now, solve for [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]
Take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{3V}{\pi h}} \][/tex]
Substitute the values we have for [tex]\( V \)[/tex], [tex]\( h \)[/tex], and [tex]\( \pi \)[/tex]:
[tex]\[ r = \sqrt{\frac{3 \times 225}{3.14 \times 8.5}} \][/tex]
After calculating, you find that the radius [tex]\( r \)[/tex] is approximately 5.03 feet when rounded to the nearest hundredth.
So, the correct formula from the options given is:
[tex]\[ r = \sqrt{\frac{3 V}{3.14 h}}, r = 5.03 \text{ feet} \][/tex]
This result matches the one provided, ensuring accuracy in determining the radius of the conical water reservoir.
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Where:
- [tex]\( V \)[/tex] is the volume
- [tex]\( r \)[/tex] is the radius of the base
- [tex]\( h \)[/tex] is the height of the cone
- [tex]\( \pi \)[/tex] is approximately 3.14
From the problem, we know:
- The volume [tex]\( V = 225 \)[/tex] cubic feet
- The height [tex]\( h = 8.5 \)[/tex] feet
Our goal is to solve for [tex]\( r \)[/tex], the radius. First, rearrange the formula to solve for [tex]\( r^2 \)[/tex]:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Multiply both sides by 3 to get rid of the fraction:
[tex]\[ 3V = \pi r^2 h \][/tex]
Now, solve for [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]
Take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{3V}{\pi h}} \][/tex]
Substitute the values we have for [tex]\( V \)[/tex], [tex]\( h \)[/tex], and [tex]\( \pi \)[/tex]:
[tex]\[ r = \sqrt{\frac{3 \times 225}{3.14 \times 8.5}} \][/tex]
After calculating, you find that the radius [tex]\( r \)[/tex] is approximately 5.03 feet when rounded to the nearest hundredth.
So, the correct formula from the options given is:
[tex]\[ r = \sqrt{\frac{3 V}{3.14 h}}, r = 5.03 \text{ feet} \][/tex]
This result matches the one provided, ensuring accuracy in determining the radius of the conical water reservoir.
