Chapter 4.2, Problem 36E

(a)

To determine

To Find: The equation for the model of the situation

Answer to Problem 36E

The require equation is $A\left(x\right)=4x^2-56x+192$ .

Explanation of Solution

Given:

The dimension of the room are 12 feet by 16 feet.

The rug should cover the $\frac{1}{2}$ of the total floor area with the uniform width that surround the rug.

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider the length of the floor is 6 ft then the width is 12 ft and the rug width is $w$ ft.

Then, the length of the rug is $l=16-2x\text{ft}$ and the width is $w=12-2x\text{ft}$ .

Then, the area of the floor is,

  $\begin{array}{c}A=16\left(12\right)\\ =192{\text{ft}}^{\text{2}}\end{array}$

Then, the area of the rug is,

  $\begin{array}{c}A\left(x\right)=\left(16-2x\right)\left(12-2x\right)\\ =192-56x+4x^2\\ =4x^2-56x+192\end{array}$

(b)

To determine

To Find: Thegraph of the model.

Answer to Problem 36E

The graph of the model is shown in Figure 1

Explanation of Solution

Consider the model is $A\left(x\right)=4x^2-56x+192$ .

Then, the graph of the model is shown in Figure 1

  

Figure 1

(c)

To determine

To Find: The dimension of the rug.

Answer to Problem 36E

The dimension of the rug is $12\text{ft}\times 8\text{ft}$ .

Explanation of Solution

Consider the area of the rug is $\frac{1}{2}$ the area of the floor.

  $\begin{array}{l}\left(16-2x\right)\left(12-2x\right)=\frac{1}{2}\left(192\right)\\ 192-56+4x^2=96\\ x^2-12x-2x+24=0\\ \left(x-12\right)\left(x-2\right)=0\end{array}$

Consider the value of $x=2$ .

Then, the length of the rug is,

  $\begin{array}{c}l=16-2x\\ =16-2\left(2\right)\\ =12\end{array}$

Then, the width is,

  $\begin{array}{c}w=12-2\left(2\right)\\ =8\end{array}$

Thus, the dimension of the rug is $12\text{ft}\times 8\text{ft}$ .