Chapter ISG, Problem 9.8.2.2P

(a)

To determine

To find: The function for the length L of the pen in term of x.

Answer to Problem 9.8.2.2P

  $L=90-2x$

Explanation of Solution

Given: C owns a goat farm and she is planning to build a new rectangular pen for 8 of her goats. She will use 90 feet of the fencing to building the other three sides of the pen.

  

Total length of the fencing is 90 feet.

According to diagram,

Perimeter of fencing $=2x+L$

  $\begin{array}{c}2x+L=90\\ L=90-2x\end{array}$

(b)

To determine

To find: The area of A of the goat pen as function of x .

Answer to Problem 9.8.2.2P

  $A=x\left(90-2x\right)$

Explanation of Solution

Given: C owns a goat farm and she is planning to build a new rectangular pen for 8 of her goats. She will use 90 feet of the fencing to building the other three sides of the pen.

  

Length of pen, $L=90-2x$

Width of pen is x

Area of pen $A=x\left(90-2x\right)$

There are two factors $x\text{ and }\left(90-2x\right)$ .

One factor shows width and one length of the pen.

(c)

To determine

To graph: The function of A on the coordinate plane.

Explanation of Solution

Given: $A=x\left(90-2x\right)$

The function of area of pen, $A=x\left(90-2x\right)$

  

(d)

To determine

To find: The maximum area of pen.

Answer to Problem 9.8.2.2P

  $\begin{array}{l}22.5\text{ feet}\\ \text{45 feet}\\ \text{1012}\text{.5 sq ft}\end{array}$

Explanation of Solution

Given: C owns a goat farm and she is planning to build a new rectangular pen for 8 of her goats. She will use 90 feet of the fencing to building the other three sides of the pen.

  

Length of pen, $L=90-2x$

Width of pen is x

Area of pen $A=x\left(90-2x\right)$

Now maximize the area of pen. Area is maximum at vertex.

  $\begin{array}{l}A=-2x^2+90x\\ x=-\frac{90}{2\left(-2\right)}=22.5\end{array}$

Put x into $L=90-2x$

  $L=90-2\left(22.5\right)=45$

Put x and L into $A=x\left(90-2x\right)$

  $A=22.5\left(90-2\left(22.5\right)\right)=1012.5$

Hence, the width of pen when area is maximum 22.5 feet and length is 45 feet.

Maximum area of pen is 1012.5 sq ft

(e)

To determine

To find: The width and length of pen if area already decided.

Answer to Problem 9.8.2.2P

  $\left(20,50\right)\text{ or }\left(25,40\right)$

Explanation of Solution

Given: C owns a goat farm and she is planning to build a new rectangular pen for 8 of her goats. She will use 90 feet of the fencing to building the other three sides of the pen.

  

Length of pen, $L=90-2x$

Width of pen is x

Area of pen $A=x\left(90-2x\right)$

Exactly area decided 1000 sq ft.

  $\begin{array}{c}x\left(90-2x\right)=1000\\ -2x^2+90x=1000\\ x^2-45x+500=0\\ \left(x-20\right)\left(x-25\right)=0\\ x=20,25\end{array}$

If $x=20$ then $L=90-2\left(20\right)=50$

If $x=25$ then $L=90-2\left(25\right)=40$