Question

1. the area of a rectangle is 84 square meters. if the length of the rectangle is 5 m longer than its width, find the dimensions. let w be the width of the rectangle in meters. 3a first find the width of the rectangle by solving for w. enter your next step here

Explanation:

Step1: Express length in terms of width

Let the width be \( w \) meters. Then the length \( l=(w + 5) \) meters.

Step2: Use the area formula

The area of a rectangle \( A=l\times w \). Given \( A = 84\), so \( w(w + 5)=84 \).
Expanding gives \( w^{2}+5w-84 = 0 \).

Step3: Factor the quadratic equation

We need to find two numbers that multiply to \(-84\) and add up to \(5\). The numbers are \(12\) and \(-7\).
So, \( w^{2}+5w - 84=(w + 12)(w-7)=0 \).

Step4: Solve for \(w\)

Set each factor equal to zero:
\( w+12 = 0\Rightarrow w=-12 \) (rejected since width cannot be negative)
\( w - 7=0\Rightarrow w = 7 \)

Answer:

The width of the rectangle is \(7\) meters.