Question

The area of a rectangular trampoline is 112 ft`^{2}`. The length of the trampoline is 6 ft greater than the width of the trampoline. This situation can be represented by the equation `w^{2}+6w-112=0.`
What is the width of the trampoline in feet?

7ft
16ft
8ft
14ft

Answer 1

To solve for the width of the trampoline, we can use the quadratic equation given:

\[
w^2 + 6w - 112 = 0
\]

We can use the quadratic formula to find the roots of the equation:

\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this equation, \(a = 1\), \(b = 6\), and \(c = -112\). Plugging in these values:

1. Calculate the discriminant:

\[
b^2 - 4ac = 6^2 - 4(1)(-112) = 36 + 448 = 484
\]

2. Calculate the square root of the discriminant:

\[
\sqrt{484} = 22
\]

3. Now plug the discriminant back into the quadratic formula:

\[
w = \frac{-6 \pm 22}{2(1)} = \frac{-6 \pm 22}{2}
\]

This gives us two possible solutions:

\[
w = \frac{16}{2} = 8 \quad \text{and} \quad w = \frac{-28}{2} = -14
\]

Since the width cannot be negative, we take \(w = 8\).

Thus, the width of the trampoline is:

\[
\boxed{8 \text{ ft}}
\]