Chapter 1.2, Problem 76E

(a)

To determine

To Find: The Volume $V$ of the rectangle as a function of $x$ and determine the domain.

Answer to Problem 76E

  $V=(108-4x)x^2$

Domain: $0<x<27$

Explanation of Solution

Given: A rectangular packet to be sent by the U.S Postal service can have maximum combined length and girth (perimeter) of $108$ inches.

  

Volume

  $\begin{array}{l}V=l\times w\times h\\ V=y\times x\times x\\ V=y{x}^2\end{array}$

But

Here

  $\begin{array}{l}y+4x=108\\ y=108-4x\end{array}$

Substitute $y=108-4x$ in volume

  $\begin{array}{l}V=y{x}^2\\ V=(108-4x)x^2\\ \mathrm{since}\\ y=108-4x>0\\ 4x<108\\ x<27\\ 0<x<27\end{array}$

Domain: $0<x<27$

(b)

To determine

To Graph: The function.

Answer to Problem 76E

  

Explanation of Solution

Given: A rectangular packet to be sent by the U.S Postal service can have maximum combined length and girth (perimeter) of $108$ inches.

  

Function: $V=(108-4x)x^2$

The graph of the function is

  

(c)

To determine

To Find: What dimensions maximize the volume of the package.

Answer to Problem 76E

  $18\times 18\times 36inches$

Explanation of Solution

Given: A rectangular packet to be sent by the U.S Postal service can have maximum combined length and girth (perimeter) of $108$ inches.

  

Function: $V=(108-4x)x^2$

The highest point on the graph is at $x=18$

Thus

The dimensions would be $18\times 18\times 36inches$