Chapter 14, Problem 36PT

(a)

To determine

To calculate: The expression that represent the cost of the carpet.

Answer to Problem 36PT

The expression for the total cost of the carper is $9x^2+90x+225$ .

Explanation of Solution

Given Information:

Width of the room is $x+5\text{feet}$ .

Cost of the carpet per square foot is $5$ .

Formula Used:

Area of a rectangle is:

  $\text{Area}=\text{Length}\times \text{Width}$

The distributive formula is:

  $\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)$

Calculation:

It is given that the length of the room is three times the width of the room.

Find the length of the room.

  $\begin{array}{c}\text{Length}=3\left(\text{Width}\right)\\ =3\left(x+5\right)\end{array}$

On the right hand side use distributive formula $\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)$

  $3\left(x+5\right)=3x+15$

Thus, length of the room is $3x+15\text{feet}$

Find the area of the room. Use the formula $\text{Area}=\text{Length}\times \text{Width}$ .

  $\text{Area}=\left(3x+15\right)\left(x+5\right)$

Find the multiplication of the expression $\left(3x+15\right)\left(x+5\right)$ .

Use distributive formula $\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)$

  $\left(3x+15\right)\left(x+5\right)=\left(3x\right)\left(x+5\right)+15\left(x+5\right)$

Use again distributiveformula $\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)$

  $\left(3x\right)\left(x+5\right)+15\left(x+5\right)=\left(3x\right)\left(x\right)+5\left(3x\right)+15\left(x\right)+15\times 5$

Use multiplication on the right hand side and simplify.

  $\begin{array}{c}\left(3x\right)\left(x+5\right)+15\left(x+5\right)=\left(3x\right)\left(x\right)+5\left(3x\right)+15\left(x\right)+15\times 5\\ =3x^2+15x+15x+75\\ =3x^2+30x+75\end{array}$

Thus, the total area of the room is $3x^2+30x+75\text{square}\text{feet}$ .

Since the cost of the carpet per square foot is $3\$$ .

Thus, find the expression for the total cost of the carpet.

  $\text{Total}\text{cost}=3\left(3x^2+30x+75\right)$

Use distributiveformula $\left(a+b\right)\left(c+d\right)=a\left(c+d\right)+b\left(c+d\right)$

  $\begin{array}{c}\text{Total}\text{cost}=3\left(3x^2+30x+75\right)\\ =9x^2+90x+225\end{array}$

Hence, the expression for the total cost of the carper is $9x^2+90x+225$ .

(b)

To determine

To calculate: The cost value of the carpet when $x$ is $5$ .

Answer to Problem 36PT

The required cost is $\$900$ .

Explanation of Solution

Given Information:

Total cost of the carpet is $9x^2+90x+225$ .

Value of $x$ is $5$ .

Calculation:

The given total cost of the carpet is $9x^2+90x+225$ .

Find the total cost when $x=5$ .

Substitute 5 for $x$ in the expression $9x^2+90x+225$ .

  $\begin{array}{c}\text{Total}\text{cost}=9x^2+90x+225\\ =9\times {\left(5\right)}^2+90\times 5+225\\ =225+450+225\\ =900\end{array}$

Hence, the required cost is $900$ .