Question

cynthia beach wants to buy a rug for a room that is 22 ft wide and 28 ft long. she wants to leave a uniform strip of floor around the rug. she can afford to buy 520 square feet of carpeting. what dimensions should the rug have? (use a comma to separate answers as needed.) the dimensions of the rug should be ft.

Explanation:

Step1: Let the width of the uniform - strip be $x$ feet.

The length of the rug is $(28 - 2x)$ feet and the width of the rug is $(22 - 2x)$ feet.

Step2: Set up the area equation.

The area of the rug is $A=(28 - 2x)(22 - 2x)=520$.
Expand the left - hand side:
\[

$$\begin{align*} (28 - 2x)(22 - 2x)&=28\times22-28\times2x-2x\times22 + 4x^{2}\\ &=616-56x - 44x+4x^{2}\\ &=4x^{2}-100x + 616 \end{align*}$$

\]
So, the equation becomes $4x^{2}-100x + 616 = 520$.

Step3: Rearrange the equation to standard quadratic form.

Subtract 520 from both sides: $4x^{2}-100x+616 - 520 = 0$, which simplifies to $4x^{2}-100x + 96 = 0$.
Divide through by 4: $x^{2}-25x + 24 = 0$.

Step4: Factor the quadratic equation.

We need two numbers that multiply to 24 and add up to 25. The numbers are 24 and 1. So, $(x - 1)(x - 24)=0$.

Step5: Solve for $x$.

Setting each factor equal to zero gives $x - 1=0$ or $x - 24=0$. So, $x = 1$ or $x = 24$. But $x = 24$ is not valid since $22-2x=22 - 48<0$ and $28-2x=28 - 48<0$. So, $x = 1$.

Step6: Find the dimensions of the rug.

The length of the rug is $28-2x=28 - 2\times1=26$ feet and the width of the rug is $22-2x=22 - 2\times1 = 20$ feet.

Answer:

20,26