Chapter 4.4, Problem 4QR

To determine

To find: The dimension of the right circular cylinder with volume $1000c{m}^3$ and surface area $600c{m}^2$ .

Answer to Problem 4QR

The dimension the right circular cylinder is $r\approx 5.419 \text{cm}$ and $r\approx 10.839 \text{cm}$ .

Explanation of Solution

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

Formula used:

The volume of the right circular cylinder is $V=\pi r^{2}h$ .

The surface area of the right circular cylinder is $A=2\pi r^2+2h\pi r$

Calculation:

The given volume of the right circular cylinder is $V=\pi r^{2}h=1000c{m}^2$ and

surface area of the right circular cylinder is $A=2\pi r^2+2h\pi r=600c{m}^2$

Implies,

  $\pi r^{2}h=1000$

Thus,

  $h=\frac{1000}{\pi r^2}$

Substitute the value of $h$ in $A=2\pi r^2+2h\pi r=600c{m}^2$ .

  $2\pi r^2+2(\frac{1000}{(\pi r^2)})\pi r=600$

Implies,

  $\begin{array}{l}A=2\pi r^2+\frac{2000}{r}\\ =600\end{array}$

Taking first derivative,

  $\frac{dA}{dx}=4\pi r-\frac{2000}{r^2}$

To find the value of $r$ assume $\frac{dA}{dx}=0$ .

Thus,

  $4\pi r-\frac{2000}{r^2}=0$

Implies,

  $4\pi r^2=2000$

Thus,

  $r=\sqrt[3]{\frac{500}{\pi }}\approx 5.419 \text{cm}$

Substituting the value of $r$ in $h=\frac{1000}{\pi r^2}$ ,

  $h=2\sqrt[3]{\frac{500}{\pi }}\approx 10.839 \text{cm}$

Thus, the dimension the right circular cylinder is $r\approx 5.419 \text{cm}$ and $r\approx 10.839 \text{cm}$ .