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6.5 polynomial equations homework
score: 16/24 answered: 16/24
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question 17
the width of a rectangle is 2 feet shorter than three times its length, and its area is 8.
if you use ( l ) to represent the length of the rectangle, then an expression representing the width is ( w=)
the length of the rectangle is feet and the width is
feet.

Explanation:

Step1: Find the expression for width

The width \(W\) is 2 feet shorter than three times the length \(L\). So, \(W = 3L-2\).

Step2: Set up the area equation

The area of a rectangle is \(A = L\times W\). Given \(A = 8\), substitute \(W = 3L - 2\) into the area formula:

$$
LATEXBLOCK0
$$

Step3: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c=0\) (\(a = 3\), \(b=-2\), \(c = - 8\)), use the quadratic formula \(L=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).

$$
LATEXBLOCK1
$$

We get two solutions for \(L\): \(L=\frac{2 + 10}{6}=2\) and \(L=\frac{2-10}{6}=-\frac{4}{3}\). Since length cannot be negative, \(L = 2\).

Step4: Find the width

Substitute \(L = 2\) into \(W=3L - 2\), then \(W=3\times2-2=4\).

Answer:

An expression representing the width is \(W = 3L-2\). The length of the rectangle is \(2\) feet and the width is \(4\) feet.