Question
You and your grandfather attend a live NBA game. One of the players is getting ready to make a free throw. The equation y=−0.05x2+0.8x+6 represents the trajectory of the ball from when it leaves the player’s hands. The player is 6 feet tall. At what distance from the player does the ball reach its maximum height?
Answers
GPT-4o mini
To find the distance from the player at which the ball reaches its maximum height, we need to determine the vertex of the parabolic equation given by \( y = -0.05x^2 + 0.8x + 6 \).
The formula for the x-coordinate of the vertex (which gives the distance where the maximum height occurs) of a parabola described by \( y = ax^2 + bx + c \) is:
\[
x = -\frac{b}{2a}
\]
In this equation:
- \( a = -0.05 \)
- \( b = 0.8 \)
Now, we can plug in the values:
\[
x = -\frac{0.8}{2(-0.05)} = -\frac{0.8}{-0.1} = 8
\]
Thus, the ball reaches its maximum height at a distance of **8 feet** from the player.
The formula for the x-coordinate of the vertex (which gives the distance where the maximum height occurs) of a parabola described by \( y = ax^2 + bx + c \) is:
\[
x = -\frac{b}{2a}
\]
In this equation:
- \( a = -0.05 \)
- \( b = 0.8 \)
Now, we can plug in the values:
\[
x = -\frac{0.8}{2(-0.05)} = -\frac{0.8}{-0.1} = 8
\]
Thus, the ball reaches its maximum height at a distance of **8 feet** from the player.