Q. 86 - Precalculus Enhanced with Graphing Utilities

Question

Constructing a Cylindrical Tube: A rectangular piece of cardboard, whose area is 216 square centimeters, is made into a cylindrical tube by joining together two sides of the rectangle. See the figure. If the tube is to have a volume of 224 cubic centimeters, what size cardboard should you start with?

Step-by-Step Solution

Answer

If the tube is to have a volume of 224 cubic centimeters, then the size cardboard should be started with is$13.02cm\times 16.59cm$.

Step 1. Finding system of equations.

Let x and y denote the dimensions of the piece of cardboard and r denote the radius of rectangular tube.

As the area of the rectangle is $216c{m}^2$, then

$xy=216$

Next the volume of the tube is $224c{m}^3$ which gives

${\mathrm{\pi r}}^2\mathrm{y}=224$.

The rectangular tube is made by joining together two sides of rectangle cardboard which gives

$$\begin{gathered} x=2\mathrm{\pi r} \\ \frac{\mathrm{x}}{2\mathrm{\pi }}=\mathrm{r} \end{gathered}$$

So, we have

$$\begin{gathered} \mathrm{\pi }{\left(\frac{\mathrm{x}}{2\mathrm{\pi }}\right)}^2\mathrm{y}=224 \\ \frac{{\mathrm{x}}^2\mathrm{y}}{4\mathrm{\pi }}=224 \end{gathered}$$

Thus the system of equations  is

$$\begin{gathered} xy=216 \\ \frac{x^{2}y}{4\mathrm{\pi }}=224 \end{gathered}$$

Step 2. Solve the system of equations.

From first equation,

$$\begin{gathered} \frac{xy}{x}=\frac{216}{x} \\ y=\frac{216}{x} \end{gathered}$$

From second equation,

$$\begin{gathered} \frac{x^{2}y}{4\mathrm{\pi }}=224 \\ \frac{x^2\left({\displaystyle \frac{216}{x}}\right)}{4\mathrm{\pi }}=224 \\ \frac{216x}{4\mathrm{\pi }}=224 \\ x=\frac{896\mathrm{\pi }}{216} \\ x=13.02cm \end{gathered}$$

and

$$\begin{gathered} y=\frac{216}{x} \\ y=\frac{216}{13.02} \\ y=16.59cm \end{gathered}$$

The required size of the cardboard is $13.02cm\times 16.59cm$.