Question

Maribel is helping the art club build backgrounds for an upcoming play. They are painting arches onto wooden planks to be behind the actors. The shape of the arches can be found by the equation y=−x2 +2x+8 . Use GeoGebra to graph this equation. What is the distance between the edges of the arches that run along the floor?(1 point)
Responses

6 feet
6 feet

4.52 feet
4.52 feet

2 feet
2 feet

8 feet

Answer 1

To find the distance between the edges of the arches along the floor, we need to determine the x-intercepts of the parabola defined by the equation \( y = -x^2 + 2x + 8 \). The x-intercepts occur where \( y = 0 \).

First, we set the equation to zero:

\[
-x^2 + 2x + 8 = 0
\]

Rearranging gives us:

\[
x^2 - 2x - 8 = 0
\]

Next, we can use the quadratic formula to find the roots of the equation, where \( a = 1 \), \( b = -2 \), and \( c = -8 \):

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

Calculating the discriminant:

\[
b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot (-8) = 4 + 32 = 36
\]

Now substituting into the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{36}}{2 \cdot 1} = \frac{2 \pm 6}{2}
\]

This gives us two possible solutions:

\[
x = \frac{8}{2} = 4 \quad \text{and} \quad x = \frac{-4}{2} = -2
\]

The x-intercepts are at \( x = 4 \) and \( x = -2 \).

Next, we calculate the distance between the edges of the arches:

\[
\text{Distance} = 4 - (-2) = 4 + 2 = 6 \text{ feet}
\]

Thus, the distance between the edges of the arches that run along the floor is **6 feet**.