Chapter 3.7, Problem 44E

a.

To determine

To write: equation of $A\left(x\right)$ .

Answer to Problem 44E

  $A\left(x\right)=\frac{480x+120}{x^2}$

Explanation of Solution

Given information:

Volume of rectangular prism = 120 cubic feet.

Length of a square = $x$

The surface area function is represented by $A\left(x\right)$

One side of the prism is fixed.

Calculation:

Let the height of rectangle be y. Look at the diagram below:

  

The volume of prism is as follows:

  $\begin{array}{c}V=\left(length\right)\left(width\right)\left(height\right)\\ =x\cdot x\cdot y\\ =x^{2}y\end{array}$

Put the volume in the above formula and solve for y as follows:

  $\begin{array}{c}120=x^{2}y\\ y=\frac{120}{x^2}\end{array}$

Since one side of the prism is fixed, so area of 5 sides will be taken to evaluate the surface area as follows:

  $\begin{array}{c}A=4xy+y\\ =4x\left(\frac{120}{x^2}\right)\text{+}\frac{120}{x^2}\left[\text{Put }y=\frac{120}{x^2}\right]\\ =\frac{480}{x}+\frac{120}{x^2}\\ =\frac{480x+120}{x^2}\end{array}$

b.

To determine

To graph: $A\left(x\right)$ .

Answer to Problem 44E

  

Explanation of Solution

Given information:

  $A\left(x\right)=\frac{480x+120}{x^2}$

Calculation:

The vertical and horizontal asymptote is $x=0$ and $y=0$ respectively. The graph for the given function is as follows:

  

c.

To determine

To find: the changes occurred in the surface area of prism for the given situation.

Answer to Problem 44E

The prism would look like a line.

Explanation of Solution

Given information:

The graph of $A\left(x\right)$ is as follows:

  

Condition for the question: $x\to 0$

Calculation:

Look for the right side of the graph only, as the surface area can’t be negative. When x get infinitely smaller the length of y gets infinitely large. So, at some point $\left(x\to 0\right)$ , the prism would look like a line at smaller value of x.