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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 only the following specifications: The height is 85 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}}{3.14h}, r = 0.56 \text{ feet}[/tex]
B. [tex]r = \frac{3\sqrt{V}}{3.14h}, r = 169 \text{ feet}[/tex]
C. [tex]r = \sqrt{\frac{3V}{3.14h}}, r = 503 \text{ feet}[/tex]
D. [tex]r = \sqrt{\frac{3V - h}{3.14}}, r = 8.22 \text{ feet}[/tex]
E. [tex]r = \sqrt{\frac{V}{3.14h}}(3), r = 871 \text{ feet}[/tex]



Answer :

GinnyAnswer
To determine the radius of a conical water reservoir given its volume [tex]\( V \)[/tex] and height [tex]\( h \)[/tex], we use the formula for the volume of a cone:

[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]

Given:
- Volume [tex]\( V = 225 \)[/tex] cubic feet
- Height [tex]\( h = 85 \)[/tex] feet

Steps to find the radius [tex]\( r \)[/tex]:

1. Start with the formula for the volume of a cone:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]

2. Rearrange the formula to solve for [tex]\( r^2 \)[/tex]:
[tex]\[ 3V = \pi r^2 h \][/tex]

3. Isolate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{3V}{\pi h} \][/tex]

4. Take the square root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{\frac{3V}{\pi h}} \][/tex]

Plug in the given values [tex]\( V = 225 \)[/tex] cubic feet and [tex]\( h = 85 \)[/tex] feet into the formula:

[tex]\[ r = \sqrt{\frac{3 \cdot 225}{3.14 \cdot 85}} \][/tex]

Proceeding with the calculations:

[tex]\[ r \approx \sqrt{\frac{675}{267.9}} \][/tex]

[tex]\[ r \approx \sqrt{2.519} \][/tex]

[tex]\[ r \approx 1.5902946558873998 \][/tex]

The radius rounded to the nearest hundredth of a foot is:

[tex]\[ r \approx 1.59 \text{ feet} \][/tex]

Therefore, the correct formula and radius are:
[tex]\[ r = \sqrt{\frac{3V}{3.14 h}} \][/tex]
[tex]\[ r = 1.59 \text{ feet} \][/tex]

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