To find the height of a cylinder when given its volume and diameter, follow these steps:
1. Identify the given data:
- Volume [tex]\( V \)[/tex] of the cylinder: [tex]\( 175\pi \)[/tex] cubic inches.
- Diameter of the cylinder: 10 inches.
2. Calculate the radius of the cylinder:
The radius [tex]\( r \)[/tex] is half of the diameter.
[tex]\[
r = \frac{\text{Diameter}}{2} = \frac{10}{2} = 5 \text{ inches}
\][/tex]
3. Recall the formula for the volume of a cylinder:
The volume [tex]\( V \)[/tex] of a cylinder is given by the formula:
[tex]\[
V = \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
4. Rearrange the formula to solve for height [tex]\( h \)[/tex]:
[tex]\[
h = \frac{V}{\pi r^2}
\][/tex]
5. Substitute the known values into the formula:
- Volume [tex]\( V = 175\pi \)[/tex]
- Radius [tex]\( r = 5 \)[/tex]
[tex]\[
h = \frac{175\pi}{\pi (5)^2}
\][/tex]
6. Simplify the expression:
- Calculate [tex]\( r^2 \)[/tex]: [tex]\( r^2 = 5^2 = 25 \)[/tex]
- Substitute into the equation:
[tex]\[
h = \frac{175\pi}{\pi \cdot 25}
\][/tex]
- Cancel out [tex]\( \pi \)[/tex] from the numerator and the denominator:
[tex]\[
h = \frac{175}{25}
\][/tex]
- Divide 175 by 25:
[tex]\[
h = 7
\][/tex]
So, the height of the cylinder is 7 inches.
