Chapter 5.4, Problem 4QR

To determine

Find the dimensions of the right circular cylinder.

Answer to Problem 4QR

two possible dimensions of the cylinder are $r=7.13cm,h=6.26cm$ and $r=4.01cm,h=19.82cm$ .

Explanation of Solution

Given information: Given volume of the cylinder is $1000c{m}^3$ and Surface of the cylinder is $600c{m}^2$ .

Calculation:

Volume of the cylinder isVolume of the cone is given by $v=\frac{1}{3}\pi r^{2}h$

Where $r,h$ are radius and height of the cylinder respectively

Substitute the values in the formula, $s=2\pi rh+2\pi r^2$

Where $r,h$ are radius and height of the cylinder respectively

Consider the volume and surface area of the cylinder.

  $\begin{array}{l}v=1000=\pi r^{2}h\\ h=\frac{1000}{\pi r^2}\end{array}$

Substitute the values of h,

  $\begin{array}{l}s=2\pi r\left(\frac{1000}{\pi r^2}\right)+2\pi r^2\\ 600=\frac{2000}{r}+2\pi r^2\\ 600r=2\pi r^3+2000\\ \pi r^3+1000-300r=0\end{array}$

Solve graphical to get, $r=-11.14,r=4.01,r=7.13$

Since radius cannot be a negative value.

  $\begin{array}{l}r=4.01,r=7.13\\ \text{here}\\ h=\frac{1000}{\pi r^2}\\ \text{For }r=4.01\\ h=\frac{1000}{\pi {\left(4.01\right)}^2}\\ =19.82\\ \text{For }r=7.13\\ h=\frac{1000}{\pi {\left(7.13\right)}^2}\\ =6.26\end{array}$

Thus, two possible dimensions of the cylinder are $r=7.13cm,h=6.26cm$ and $r=4.01cm,h=19.82cm$ .