Chapter 1.4, Problem 66E

i.

To determine

To calculate: A rectangular package has a combined length and girth (perimeter of cross section) of 108 inches. Write the volume V of the package as a function of x. What is the domain of the function?

  

Answer to Problem 66E

Volume V of the package is $108x^2-4x^3$ and domain $x\in \left(0,27\right)$

Explanation of Solution

Given information: Combined length and girth (perimeter of cross section) of a rectangular package is 108 inches. so

  $\begin{array}{l}108=y+4x\\ y=108-4x\cdots \left(i\right)\end{array}$

Formula used: $V=l\times b\times h$

Calculation:

Here length is x, breadth is x and height is y , so

  $V=x\times x\times y=x^{2}y\cdots \left(ii\right)$

Put the value of y from equation $\left(i\right)$ in equation $\left(ii\right)$ , we get

  $\begin{array}{c}V=x^2\left(108-4x\right)\\ =108x^2-4x^3\end{array}$

Domain is the set of all values for which the function is defined.

Volume is always greater than zero. i.e.

  $\begin{array}{l}108x^2-4x^3>0\\ \left(27-x\right)4x^3>0\\ \Rightarrow x>0\text{or}27-x>0\\ \Rightarrow x>0\text{or}27>x\\ \Rightarrow 0<x<27\end{array}$

Conclusion: Volume V of the package is $108x^2-4x$ and domain $x\in \left(0,27\right)$

ii.

To determine

To graph: A rectangular package has a combined length and girth (perimeter of cross section) of 108 inches Use a graphi.cal utility to graph the function.

Answer to Problem 66E

  

Explanation of Solution

Given information:

From the above part $V=x^2\left(108-4x\right)$

Concept used:

To plot the graph keep $V=0$

  $\begin{array}{l}{x}^2\left(108-4x\right)=0\\ \Rightarrow x=0\text{or}108-4x=0\\ \Rightarrow x=0\text{or}108=4x\\ \Rightarrow x=0\text{or}x=27\end{array}$

So, when $y=0,x=0\text{or}x=27$

iii.

To determine

To calculate: A rectangular package has a combined length and girth (perimeter of cross section) of 108 inches. What dimensions will maximize the volume of the package? Explain

Answer to Problem 66E

The dimensions are $18\text{in}\times 18\text{in}\times 36\text{in}$

Explanation of Solution

Given information: From the above graph we can see that volume is maximized when $x=18$

  $\begin{array}{c}y=108-4x\\ =108-4\times 18\\ =36\end{array}$

so length = 18 inches, width = 18 inches and height = 36 inches

Conclusion: The dimensions are $18\text{in}\times 18\text{in}\times 36\text{in}$