Question

algebra skill review #2
score: 11/21 answered: 11/21
question 12
the length of a rectangle is 3 feet more than 2 times the width. if the area is 14 square feet, find the width and the length.
width: feet
length: feet
question help: ebook message instructor

Explanation:

Step1: Define variables

Let the width of the rectangle be \( w \) feet. Then the length \( l=(2w + 3) \) feet.

Step2: Use the area formula

The area formula for a rectangle is \( A=l\times w \). Given \( A = 14 \), we substitute \( l \) and \( w \) into the formula: \( 14=(2w + 3)\times w \).
Expanding gives \( 2w^{2}+3w-14 = 0 \).

Step3: Solve the quadratic equation

For a quadratic equation \( ax^{2}+bx + c=0 \) (here \( a = 2 \), \( b = 3 \), \( c=-14 \)), we use the quadratic formula \( w=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \).
First, calculate the discriminant \( \Delta=b^{2}-4ac=(3)^{2}-4\times2\times(-14)=9 + 112=121 \).
Then \( w=\frac{-3\pm\sqrt{121}}{4}=\frac{-3\pm11}{4} \).
We have two solutions for \( w \):
\( w_{1}=\frac{-3 + 11}{4}=\frac{8}{4}=2 \) and \( w_{2}=\frac{-3-11}{4}=\frac{-14}{4}=-\frac{7}{2} \).
Since width cannot be negative, \( w = 2 \) feet.

Step4: Find the length

Substitute \( w = 2 \) into \( l=2w + 3 \). Then \( l=2\times2+3=4 + 3=7 \) feet.

Answer:

Width: \( 2 \) feet
Length: \( 7 \) feet