Answer
$$V_1= \frac{l^2}{4\pi} W$$
$$V_2= \frac{W^2}{4\pi} l$$
Work Step by Step
Generally, we know that the volume of a cylinder is defined as:
$V=\pi r^2h$
We have to define $r$, and we will use the formula of circumference for that:
$C=2\pi r$
$(1)$Let's first calculate the first scenario, where:
$Circumference(C) = l$
$Height(h)=W$
Define $r$ in terms of the circumference
$l=2\pi r$
$r=\frac{l}{2\pi}$
So we get the following:
$V_1=\pi (\frac{l}{2\pi})^2 W$
Simplify:
$V_1= \frac{l^2}{4\pi} W$
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$(1)$Let's calculate the second scenario, where:
$Circumference(C) = W$
$Height(h)=l$
Define $r$ in terms of the circumference
$W=2\pi r$
$r=\frac{W}{2\pi}$
So we get the following:
$V_2=\pi (\frac{W}{2\pi})^2 l$
Simplify:
$V_2= \frac{W^2}{4\pi} l$
