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=−x^2 +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

8 feet
8 feet

4.52 feet
4.52 feet

2 feet
2 feet

6 feet

Answer 1

To determine the distance between the edges of the arches that run along the floor, we need to find the x-intercepts of the quadratic equation given by \( y = -x^2 + 2x + 8 \). The x-intercepts are found by setting \( y = 0 \).

The equation becomes:
\[
0 = -x^2 + 2x + 8
\]

Rearranging gives us:
\[
x^2 - 2x - 8 = 0
\]

Next, we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = -2 \), and \( c = -8 \).

Calculating the discriminant:
\[
b^2 - 4ac = (-2)^2 - 4(1)(-8) = 4 + 32 = 36
\]

Now we can find the x-intercepts:
\[
x = \frac{-(-2) \pm \sqrt{36}}{2(1)} = \frac{2 \pm 6}{2}
\]

This will give us two solutions:
1. \( x = \frac{2 + 6}{2} = \frac{8}{2} = 4 \)
2. \( x = \frac{2 - 6}{2} = \frac{-4}{2} = -2 \)

The x-intercepts are at \( x = 4 \) and \( x = -2 \). Now to find the distance between these two points on the x-axis, we calculate:
\[
\text{Distance} = 4 - (-2) = 4 + 2 = 6
\]

Thus, the distance between the edges of the arches that run along the floor is:

**6 feet**