Question

the length of a rectangle is 4 more than the width. the area of the rectangle is 60 square yards. what is the length of the rectangle? let w = width and 4 + w = length. which equation represents the situation? done

Explanation:

Step1: Recall the area formula for a rectangle

The area \( A \) of a rectangle is given by the formula \( A=\text{length}\times\text{width} \).

Step2: Substitute the given expressions for length and width

We know that the width is \( w \) and the length is \( w + 4 \), and the area \( A = 60 \). Substituting these into the area formula, we get \( w(w + 4)=60 \).

Step3: Expand and solve the quadratic equation

Expanding the left - hand side, we have \( w^{2}+4w = 60 \), which can be rewritten as \( w^{2}+4w-60 = 0 \).
We can solve this quadratic equation by factoring. We need to find two numbers that multiply to \( - 60 \) and add up to \( 4 \). The numbers are \( 10 \) and \( - 6 \).
So, \( w^{2}+4w - 60=(w + 10)(w - 6)=0 \).
Setting each factor equal to zero gives \( w+10 = 0 \) or \( w - 6=0 \).
Solving for \( w \), we get \( w=-10 \) or \( w = 6 \). Since the width cannot be negative, we take \( w = 6 \).

Step4: Find the length

The length is given by \( l=w + 4 \). Substituting \( w = 6 \) into this expression, we get \( l=6 + 4=10 \).

Answer:

The length of the rectangle is 10 yards. The equation that represents the situation is \( w(w + 4)=60 \) (or \( w^{2}+4w-60 = 0 \)).