Q. 325 - Intermediate Algebra

Question

The area of a rectangular carpet is $28$ square feet. The length is three feet more than the width. Find the length and the width of the carpet.

Step-by-Step Solution

Answer

The length and the width of the rectangular carpet is $4 feet, 7 feet$.

Step 1. Assumption of integers.

Assume $x feet$ to be the width and $x+3 feet$ to be the length of the carpet.

Area of rectangular carpet$=length\times width$

Substitute the values in the equation of area,

$$\begin{gathered} 28=x\left(x+3\right) \\ {x}^2+3x=28 \\ {x}^2+3x-28=0 ...... \left(i\right) \end{gathered}$$

Step 2. Factor the equation.

On factoring,

$$\begin{gathered} x^2+7x-4x-28=0 \\ x\left(x+7\right)-4\left(x+7\right)=0 \\ \left(x+7\right)\left(x-4\right)=0 \end{gathered}$$

Use the zero product property,

$$\begin{gathered} x+7=0 \\ x=-7 \end{gathered}$$

Then,

$$\begin{gathered} x-4=0 \\ x=4 \end{gathered}$$

As we know, the width of the carpet cannot be negative, so $x=4$.

Then the length of the carpet,

$$\begin{gathered} =x+3 \\ =4+3 \\ =7 \end{gathered}$$

Step 3. Check the answers.

Substitute the width and the length in equation (i),

$$\begin{gathered} 4\times 7=28 \\ 28=28 \end{gathered}$$

This is true.

Hence, the width and the length of the rectangular carpet is $4 feet, 7 feet$.