Question

Terry throws a ball into a hoop that is 5 feet off the ground. The path of the ball can be modeled by the equation
y
=
−
x
2
+
4
x
+
5
, where x represents the horizontal distance traveled by the ball, in feet, and y represents the vertical height of the ball, in feet. Since the ball hoop is located 5 feet off the ground, you can use the equation
y
=
5
to model the position of the hoop. Determine the horizontal distance between Terry and the hoop.(1 point)
Responses

5 feet
5 feet

4 feet
4 feet

2 feet
2 feet

0 feet

Answer 1

To determine the horizontal distance between Terry and the hoop, we need to find the points where the path of the ball (given by the equation \( y = -x^2 + 4x + 5 \)) intersects the height of the hoop \( y = 5 \).

We can set the two equations equal to find the values of \( x \):

\[
-x^2 + 4x + 5 = 5
\]

Subtract 5 from both sides:

\[
-x^2 + 4x = 0
\]

Factoring out \( x \):

\[
x(-x + 4) = 0
\]

Setting each factor to zero gives us:

1. \( x = 0 \)
2. \( -x + 4 = 0 \) ⇒ \( x = 4 \)

So, the ball is at the height of the hoop at \( x = 0 \) feet and \( x = 4 \) feet.

Terry throws the ball from \( x = 0 \) to the hoop which is at \( x = 4 \). Thus, the horizontal distance between Terry and the hoop is:

\[
4 - 0 = 4 \text{ feet}
\]

Therefore, the answer is **4 feet**.