Question

65-70 Find a formula for the described function and state its domain.

70. A right circular cylinder has volume \(25{{\mathop{\rm in}\nolimits} ^3}\). Express the radius of the cylinder as a function of the height.

Short answer

The formula for the described function is \(f\left( h \right) = \frac{5}{{\sqrt {\pi h} }}{\mathop{\rm in}\nolimits} \), and the domain of \(f\) is \(h > 0\).

Step-by-step solution

Define the domain of the function

The domain of the function is the set of all inputs for which the formula produces a real number output.

Determine the formula for the described function and state its domain

Consider \(r\) and \(h\) as the radius and height of the right circular cylinder.

The volume \(V\) of the right circular cylinder is calculated below:

\(\begin{aligned}V &= \pi {r^2}h\\25 &= \pi {r^2}h\\{r^2} &= \frac{{25}}{{\pi h}}\end{aligned}\)

Solve for radius \(r\) and reject the negative solution to yield \(r = \frac{5}{{\sqrt {\pi h} }}\).

Express the radius of the cylinder as the function of height.

\(f\left( h \right) = \frac{5}{{\sqrt {\pi h} }}{\mathop{\rm in}\nolimits} \)

Thus, the formula for the described function is \(f\left( h \right) = \frac{5}{{\sqrt {\pi h} }}{\mathop{\rm in}\nolimits} \), and the domain of \(f\) is \(h > 0\).