Question

6.5 polynomial equations homework
score: 17/24 answered: 17/24
question 18
the width of a rectangle is 2 feet longer than three times its length, and its area is 85.
if you use ( l ) to represent the length of the rectangle, then an expression representing the width is ( w=)
the area of the rectangle is 85 square feet.
an equation for the area of the rectangle in terms of the length ( l ) is:
the length of the rectangle is feet and the width is
feet.

Explanation:

Step1: Find the expression for width

The width is 2 feet longer than three times the length. If length is \(L\), then width \(W = 3L+2\).

Step2: Form the area equation

The area of a rectangle is \(A = L\times W\). Substituting \(W = 3L + 2\) and \(A=85\), we get \(L(3L + 2)=85\). Expanding, \(3L^{2}+2L-85 = 0\).

Step3: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c=0\) (\(a = 3\), \(b = 2\), \(c=-85\)), use the quadratic formula \(L=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).

$$
LATEXBLOCK0
$$

We have two solutions for \(L\): \(L=\frac{-2 + 32}{6}=\frac{30}{6}=5\) and \(L=\frac{-2-32}{6}=\frac{-34}{6}\) (reject since length cannot be negative).

Step4: Find the width

Substitute \(L = 5\) into \(W=3L + 2\), \(W=3\times5+2=15 + 2=17\).

Answer:

• \(W=3L + 2\)
• \(3L^{2}+2L-85 = 0\)
• Length \(L = 5\) feet, Width \(W = 17\) feet.