Question

the area of a rectangle is $21\\ \text{in}^2$. the length is 2 times the width plus 1. use $w$ for width. what is the length in inches? equation?

Explanation:

Step1: Define variables and length expression

Let the width be \( w \) inches. The length \( l \) is \( 2w + 1 \) inches (since length is 2 times width plus 1). The area of a rectangle is given by \( A = l \times w \), and the area \( A = 21 \) \( \text{in}^2 \). So the equation is \( w(2w + 1)=21 \).

Step2: Expand and form quadratic equation

Expand the left - hand side: \( 2w^{2}+w = 21 \). Then, rearrange it to the standard quadratic form \( 2w^{2}+w - 21=0 \).

Step3: Solve the quadratic equation

We can solve the quadratic equation \( 2w^{2}+w - 21 = 0 \) using the quadratic formula \( w=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \), where \( a = 2 \), \( b = 1 \), and \( c=- 21 \).

First, calculate the discriminant \( \Delta=b^{2}-4ac=(1)^{2}-4\times2\times(-21)=1 + 168 = 169 \).

Then, \( w=\frac{-1\pm\sqrt{169}}{2\times2}=\frac{-1\pm13}{4} \).

We have two solutions for \( w \):

• \( w=\frac{-1 + 13}{4}=\frac{12}{4}=3 \)
• \( w=\frac{-1-13}{4}=\frac{-14}{4}=-3.5 \)

Since the width cannot be negative, we take \( w = 3 \) inches.

Step4: Calculate the length

Now that we know \( w = 3 \), substitute \( w \) into the expression for length \( l=2w + 1 \).

\( l=2\times3+1=6 + 1=7 \) inches.

Answer:

The length of the rectangle is 7 inches.