Chapter 4, Problem 87RE

(a)

To determine

To express: the amount $A$ of material to make the can

Answer to Problem 87RE

The amount of material required can be expressed as

  $A=2\pi r^2+\frac{500}{r}$

Explanation of Solution

Given:

volume = 250 cubic centimetres

Calculation:

So, let us take

Radius of the cylindrical can $=r\text{cm}$

Height of the cylindrical can $=h\text{cm}$

Now

Volume of the cylindrical can $=\pi r^{2}h$

Here the volume as 250 cubic centimeters. So, we get

  $\pi r^{2}h=250$

  $h=\frac{250}{\pi r^2}\dots \dots (1)$

Here, express the amount $A$ of the material required to make the can in terms of $r$

Now, the amount of material required to make the can is equal to the total surface area of the can. So,

  $A=2\pi r^2+2\pi rh\cdots \cdots (2)$

Since, we have to express the amount in terms of $r,$ we have to eliminate $h$ from the equation.

Using the value of $h$ from equation (1) in (2) , we get.

  $A=2\pi r^2+2\pi r\left(\frac{250}{\pi r^2}\right)$

  $A=2\pi r^2+\frac{500}{r}$

Therefore, the amount of material required can be expressed as

  $A=2\pi r^2+\frac{500}{r}$

Conclusion:

Therefore, the amount of material required can be expressed as

  $A=2\pi r^2+\frac{500}{r}$

(b)

To determine

To find:the amount of material required

Answer to Problem 87RE

The amount of material required is $223.19{\text{cm}}^2$

Explanation of Solution

Calculation:

Here, find out the amount of material required if the radius of the can is $3\text{cm}$ .

So, using $r=3$ in equation $(3),$

  $A=2\left(\frac{22}{7}\right){(3)}^2+\frac{500}{3}$

  $A=56.52+166.66$

  $A=223.19$

Therefore, the amount of material required is $223.19{\text{cm}}^2$

Conclusion:

Therefore,the amount of material required is $223.19{\text{cm}}^2$

(c)

To determine

To find:the amount of material required if radius is 5 centimetres

Answer to Problem 87RE

Therefore, the amount of material required is $257{\text{cm}}^2$

Explanation of Solution

Calculation:

Here, find out the amount of material required if the radius of the can is $5\text{cm}$ .

So, using $r=5$ in equation $(3),$ we get,

  $A=2\left(\frac{22}{7}\right){(5)}^2+\frac{500}{5}$

  $A=157+100$

  $A=257$

Therefore,the amount of material required is $257{\text{cm}}^2$

Conclusion:

Therefore,the amount of material required is $257{\text{cm}}^2$

(d)

To determine

To graph: $A=A(r)$

Answer to Problem 87RE

The amount of material required $A$ would be smallest.

Explanation of Solution

Calculation:

Here, we need to plot a graph for $A$ as a function of r. So, use the graphing utilitysoftware to plot the graph.

  $A=2\pi r^2+\frac{500}{r}$

  

From the graph, the value of $A$ will the lowest when $r=3.41\text{cm}$

Therefore,

For $\sqrt{r=3.41\text{cm}}$ the amount of material required $A$ would be smallest.

Conclusion:

Therefore,

For $r=3.41\text{cm}$ the amount of material required $A$ would be smallest.