College

You work for a store that sells built-to-order water reservoirs. Your manager asks you to visit a small business to measure a damaged conical water reservoir for replacement. The label on the water reservoir indicates the following specifications:

The height is 8.5 feet, and when full, the water reservoir holds 225 cubic feet of water.

Which formula will determine the radius of the water reservoir? Rounded to the nearest hundredth of a foot, what is the radius of the water reservoir?

A. [tex]r = \frac{\sqrt{V}}{3.14 h}, r = 0.56[/tex] feet
B. [tex]r = \frac{3 \sqrt{\nabla}}{3.14 h}, r = 1.69[/tex] feet
C. [tex]r = \sqrt{\frac{3 V}{3.14 h}}, r = 5.03[/tex] feet
D. [tex]r = \sqrt{\frac{3 V - \pi}{3.14}}, r = 8.22[/tex] feet
E. [tex]r = \sqrt{\frac{V}{3.14 h}}(3), r = 8.71[/tex] feet

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.