Chapter 4, Problem 46RE

(a)

To determine

To express the amount A of the material to make the can.

Answer to Problem 46RE

The amount of material $A=2\pi r^2+\frac{500}{r}$

Explanation of Solution

Given:

The Volume of the right circular cylinder is 250 cubic centimeters.

Concept Used:

Volume of the right circular cylinder:

  $V=\pi r^{2}h$

Area of the right circular cylinder:

Calculation:

We have ,

  $\begin{array}{l}V=250c{m}^3\\ 250c{m}^3=\pi r^{2}h\end{array}$

  $\begin{array}{l}=\pi r^{2}h=250\\ =h=\frac{250}{\pi r^{2}h}\end{array}$

Now,

  $A=2\pi r^2+2\pi rh$

  $\begin{array}{l}=2\pi r^2+2\pi r\left(\frac{250}{\pi r^2}\right)\\ =2\pi r^2+\frac{2(250)}{r}\\ =2\pi r^2+\frac{500}{r}\end{array}$

Conclusion:

The amount $A=2\pi r^2+\frac{500}{r}$

(b)

To determine

If the can is of radius 3 centimeters, how much material is required?

Answer to Problem 46RE

The amount of material $A=18\pi +166.6666$ .

Explanation of Solution

Given:

  $r=3cm$

Concept Used:

  $A=2\pi r^2+\frac{500}{r}$

Calculation:

We have,

  $r=3cm$

Therefore,

  

  $\begin{array}{l}$ =2\pi {(3)}^2+\frac{500}{3}$=2\pi (9)+\frac{500}{3}\\ =18\pi +166.6666\end{array}$

Conclusion:

The amount $A=18\pi +166.6666$ .

(c)

To determine

If the can is of radius 5 centimeters, how much material is required?

Answer to Problem 46RE

The amount of material $A=50\pi +100$ .

Explanation of Solution

Given:

  $r=5cm$

Concept Used:

  $A=2\pi r^2+\frac{500}{r}$

Calculation:

We have,

  $r=5cm$

Therefore,

  

  $A=2\pi r^2+\frac{500}{r}$

  $\begin{array}{l}$ =2\pi {(5)}^2+\frac{500}{5}$=2\pi (25)+\frac{500}{5}\\ =50\pi +100\end{array}$

Conclusion:

The amount $A=50\pi +100$ .

(d)

To determine

To calculate: The value of r for which A is smallest .

Answer to Problem 46RE

The value of r at which $A(r)$ is smallest is 3. With the following graph:

  

Explanation of Solution

Given information:

The amount of material $A=2\pi r^2+\frac{500}{r}$ .

Formula used:

  $A=2\pi r^2+\frac{500}{r}$

Calculation:

It is provided that $A=2\pi r^2+\frac{500}{r}$ .

Re-writing the equation as :-

  $y=2\pi x^2+\frac{500}{x}$

Substitute the values of $x$ in the equation $y=2\pi x^2+\frac{500}{x}$ we have,

$x$	$y=2\pi x^2+\frac{500}{x}$
$0$	$undefined$
$1$	$506.28319$
$2$	$275.13274$
$3$	$223.21533$
$4$	$225.53096$
$5$	$257.07963$
$6$	$309.528$
$7$	$379.30465$

By using the above table a graph has been drawn. Also, the table shows that at

  $\begin{array}{l}=r=3\\ =A(r)=223.21533\end{array}$

the smallest value among all the values because with after that the value of $A(r)$ has been increasing continuously.

  

Thus, with the different value of r the graph $A=A(r)$ is represented above. With an increase in r the value of $A(r)$ except $r=0$ the amount of material is undefined.