Question

in exercises 15–20, write the specified quantity as a function of the specified variable. it will help in each case to draw a picture. 15. the height of a right - circular cylinder equals its diameter. write the volume of the cylinder as a function of its radius.

Explanation:

Step1: Recall volume formula

The volume formula of a right - circular cylinder is $V=\pi r^{2}h$, where $r$ is the radius and $h$ is the height.

Step2: Express height in terms of radius

Given that the height $h$ equals the diameter $d$, and since $d = 2r$, then $h=2r$.

Step3: Substitute height into volume formula

Substitute $h = 2r$ into $V=\pi r^{2}h$. We get $V=\pi r^{2}(2r)$.

Step4: Simplify the function

$V = 2\pi r^{3}$.

Answer:

$V(r)=2\pi r^{3}$