Chapter 9.9, Problem 40SR

(a)

To determine

To calculate: To write and graph the equation to represent the area $A$ of the pen.

Answer to Problem 40SR

Equation to represent the area $A$ of the pen is $A=x\left(50-x\right)$

Explanation of Solution

Given information: A dog pen is being built with a $100$ foot roll of chain link fence

  

Formula Used:

Perimeter of rectangle is given as

  $Perimeter=2\left(length+width\right)$

Area of rectangle is given as

  $Area=length\times width$

Calculation:

Given that a dog pen is being built with a $100$ foot roll of chain link fence as shown below

  

Equation to represent the area $A$ of the pen is given as

  $\begin{array}{l}Area=length\times width\\ A=x\left(50-x\right)\end{array}$

In order to graph the equation, find the solutions of the equation

When $x=0$

  $\begin{array}{l}A=0\left(50-0\right)\\ A=0\end{array}$

When $x=1$

  $\begin{array}{l}A=1\left(50-1\right)\\ A=49\end{array}$

When $x=10$

  $\begin{array}{l}A=10\left(50-10\right)\\ A=10\left(40\right)\\ A=400\end{array}$

When $x=20$

  $\begin{array}{l}A=20\left(50-20\right)\\ A=20\left(30\right)\\ A=600\end{array}$

Plotting the points on graph:

  

Conclusion:

Hence, equation to represent the area $A$ of the pen is $A=x\left(50-x\right)$

(b)

To determine

To calculate: To find the dimensions of the dog pen that would enclose the maximum area inside the fence

Answer to Problem 40SR

The dimensions of the dog pen that would enclose the maximum area inside the fence are $25foot\times 25foot$

Explanation of Solution

Given information: A dog pen is being built with a $100$ foot roll of chain link fence

  

Calculation:

Given that a dog pen is being built with a $100$ foot roll of chain link fence as shown below

  

Equation to represent the area $A$ of the pen is given as

  $\begin{array}{l}Area=length\times width\\ A=x\left(50-x\right)\end{array}$

Graph of the function is as follows:

  

From the graph, the maximum point is at $x=25$

Thus, when dimension of dog pen is $25foot\times 25foot$ , then the dog pen would enclose the maximum area inside the fence.

Conclusion:

Hence, dimensions of the dog pen that would enclose the maximum area inside the fence are $25foot\times 25foot$