Q 6.106 - Intermediate Algebra

Question

A rectangular patio has area $180$ square feet. The width of the patio is three feet less than the length. Find the

length and width of the patio.

Step-by-Step Solution

Answer

The length of the rectangular patio is $15 feet$ and the width of the rectangular patio is $12 feet.$

Step 1. Given Information

There is a rectangular patio whose area is $180$ square feet and the width of the patio is three feet less than the length.

We have to find the length and width of the patio.

Step 2. Assume the length and width of the patio

Let the length of the patio be x

As we know the width of the patio is three feet less than the length,

So the width of the patio will be $x-3.$

Step 3. Summarize into an equation

The area of the rectangular patio is $180$ square feet.

By using the formula of area of a rectangle

$$\begin{gathered} \left(x\right)(x-3)=180 \\ {x}^2-3x-180=0 \\ {x}^2-15x+12x-180=0 \\ x(x-15)+12(x-15)=0 \\ (x-15)(x+12)=0 \end{gathered}$$

Step 4. Use the Zero Product Property

The equation we get is

$(x-15)(x+12)=0$

So,

$$\begin{gathered} x+12=0 \\ x=-12 \end{gathered}$$ and $$\begin{gathered} x-15=0 \\ x=15 \end{gathered}$$

Step 5. Substitute the value of x in assumed width and length to find the exact width and length

Since x is the length of the rectangular patio, it does not be negative therefore, we will eliminate the value $-12.$

If $x=15$

The length of the patio will be $15 feet.$

The width of the patio will be $15-3=12 feet.$