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]
