### Instruction: A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars will increase sales. If the diameter of the jars is increased by $25\%$ without altering the volume, by what percent must the height be decreased?### Response: I need to find the percentage change in the height of the jar when the volume stays constant and the diameter increases by $25\%$.
The volume of a cylinder is given by $V = \pi r^2 h$, where $r$ is the radius and $h$ is the height.
If the diameter increases by $25\%$, then the radius also increases by $25\%$.
That means the new radius is $1.25$ times the original radius, or $r' = 1.25 r$.
Since the volume is constant, I can set the old and new volumes equal to each other and solve for the new height, $h'$.
$V = \pi r^2 h = \pi r'^2 h'$
Substituting $r' = 1.25 r$, I get
$\pi r^2 h = \pi (1.25 r)^2 h'$
Simplifying, I get
$r^2 h = 1.25^2 r^2 h'$
Dividing by $r^2$, I get
$h = 1.25^2 h'$
Solving for $h'$, I get
$h' = \frac{h}{1.25^2}$
Now I need to find the percentage decrease in the height, which is the difference between the old and new heights divided by the old height, times $100\%$.
$\text{Percent decrease} = \frac{h - h'}{h} \times 100\%$
Substituting $h' = \frac{h}{1.25^2}$, I get
$\text{Percent decrease} = \frac{h - \frac{h}{1.25^2}}{h} \times 100\%$
Simplifying, I get
$\text{Percent decrease} = \frac{1 - \frac{1}{1.25^2}}{1} \times 100\%$
Evaluating, I get