Chapter 6.5, Problem 78PPS

a.

To determine

To draw:a diagram to represent the situation.

Answer to Problem 78PPS

  $A=5760\text{sq}\text{feet}$ .

Explanation of Solution

Given:

Frodo’s corral, an enclosure for livestock is currently $32\text{ft}\text{by }40\text{ft}$ .

Calculation:

As the enclosure for livestock is currently $32\text{ft}\text{by }40\text{ft}$ .

Now, the area of enclosure was to enlarge by $4.5\text{times}$ .

Therefore, the pictorial representation of this situation is

  

Hence, the area of new rectangular corral is $A=5760\text{sq}\text{feet}$ .

b.

To determine

To write: the polynomial equation for the area of new corral and find the factor of the equation.

Answer to Problem 78PPS

  $x=40$ .

Explanation of Solution

Given:

The area of new corral is $A=5760\text{sq}\text{feet}$ .

Calculation:

Let x be the increase in length and width of the corral. Now, make an equation for the given condition. Therefore, according to question

  $\begin{array}{l}\left(x+40\right)\left(x+32\right)=1280\left(4.5\right)\left\{\text{given condition}\right\}\\ x^2+72x+1280=5760\left\{expand\right\}\\ x^2+72x-4480=0\left\{\text{subtract 5760 from both sides}\right\}\\ \left(x-40\right)\left(x+112\right)=0\{\text{factor}}\end{array}$

Use zero product property, then

  $\begin{array}{l}\left(x-40\right)=0\text{ or }\left(x+112\right)=0\\ \Rightarrow x=40\text{or }x=-112\end{array}$

Since, the dimensions cannot be negative, therefore, $x=40$ .

Hence, the factor of that equation is $x=40$ .

c.

To determine

To graph:the given function.

Answer to Problem 78PPS

The graph is shown in figure.

Explanation of Solution

Given:

The given polynomial is $A\left(d\right)=d^2+72d-4480$ .

Calculation:

From part (b), $A\left(d\right)=d^2+72d-4480$ .

Let $d=2x$ , then

  $\begin{array}{l}A\left(x\right)=4x^2+144x-4480\\ A\left(x\right)=4\left(x-20\right)\left(x+56\right)\left\{\text{factor}\right\}\end{array}$

Now, by zero product property

  $\begin{array}{l}x-20=0\Rightarrow x=20\\ x+56=0\Rightarrow x=-56\\ x_V=-\frac{144}{2.4}=-18\left\{\text{vertex}\right\}\end{array}$ .

Thus, the y -intercept is $y=4{\left(0\right)}^2+144\left(0\right)-4480=-4480$

Now, the graph of the given function is

  

d.

To determine

To explain: that for any value of x is irrelevant.

Answer to Problem 78PPS

  $x=-112$ is irrelevant.

Explanation of Solution

Given:

The given polynomial is $x^2+72x-4480=0$ .

Calculation:

From part (b), there are two value of x which is $x=40,x=-112$ .

Since, the dimensions of the corral cannot be negative,

So $x=-112$ is irrelevant.

Hence, the value of $x=-112$ is irrelevant.