Question

practice 7 (from unit 2, lesson 9)
the volume of a cylinder is represented by the formula $v = \pi r^2 h$
find each missing height and show your reasoning.
type your answers in the boxes.

volume (cubic inches)	radius (inches)	height (inches)
$31.25\pi$	2.5	\boxed{}
$v$	$r$	\boxed{}

Explanation:

Step1: Solve for height formula

Given \( V = \pi r^2 h \), solve for \( h \): divide both sides by \( \pi r^2 \), so \( h=\frac{V}{\pi r^2} \).

Step2: First row calculation

For \( V = 96\pi \), \( r = 4 \):
Substitute into \( h=\frac{V}{\pi r^2} \), get \( h=\frac{96\pi}{\pi \times 4^2} \).
Simplify: \( \pi \) cancels, \( 4^2 = 16 \), so \( h=\frac{96}{16}=6 \).

Step3: Second row calculation

For \( V = 31.25\pi \), \( r = 2.5 \):
Substitute into \( h=\frac{V}{\pi r^2} \), get \( h=\frac{31.25\pi}{\pi \times (2.5)^2} \).
Simplify: \( \pi \) cancels, \( (2.5)^2 = 6.25 \), so \( h=\frac{31.25}{6.25}=5 \).

Step4: General formula

For \( V \) and \( r \), \( h=\frac{V}{\pi r^2} \) (or \( h=\frac{V}{\pi r^2} \) simplifies to \( h=\frac{V}{\pi r^2} \), which can also be written as \( h=\frac{V}{\pi r^2} \), or factoring out \( \pi \) and \( r^2 \), it's \( h=\frac{V}{\pi r^2} \), but more neatly, from \( V = \pi r^2 h \), solving for \( h \) gives \( h=\frac{V}{\pi r^2} \), which is the same as \( h=\frac{V}{\pi r^2} \), or if we divide numerator and denominator by \( \pi \), \( h=\frac{V/\pi}{r^2} \), but the key is the formula \( h = \frac{V}{\pi r^2} \)).

Answer:

First height: \( 6 \)
Second height: \( 5 \)
General height formula: \( \boldsymbol{h = \frac{V}{\pi r^2}} \) (or \( \boldsymbol{\frac{V}{\pi r^2}} \))