Question

An art class is painting arches onto wooden planks. The shape of the arches an be found by the equation y = -x^2 +2x +8. What is the distance between the edges of arches

Answer 1

To find the distance between the edges of the arches defined by the quadratic equation \( y = -x^2 + 2x + 8 \), we need to first determine the x-coordinates where the arches intersect the x-axis (i.e., where \( y = 0 \)). This occurs when:

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

We can rearrange this to:

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

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 applying the quadratic formula:

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

This gives us two solutions:

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

The edges of the arches are at \( x = -2 \) and \( x = 4 \). To find the distance between these points on the x-axis:

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

Thus, the distance between the edges of the arches is \( 6 \) units.