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
𝑦
=
−
0.05
𝑥
2
+
0.8
𝑥
+
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?
𝑦
=
−
0.05
𝑥
2
+
0.8
𝑥
+
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 x-coordinate of the vertex of a parabola described by the equation \(y = ax^2 + bx + c\) can be calculated using the formula:
\[
x = -\frac{b}{2a}
\]
In this case, \(a = -0.05\) and \(b = 0.8\). Plugging these values into the formula gives:
\[
x = -\frac{0.8}{2 \cdot -0.05} = -\frac{0.8}{-0.1} = 8
\]
Thus, the ball reaches its maximum height at a distance of \(8\) feet from the player.
\[
y = -0.05x^2 + 0.8x + 6
\]
The x-coordinate of the vertex of a parabola described by the equation \(y = ax^2 + bx + c\) can be calculated using the formula:
\[
x = -\frac{b}{2a}
\]
In this case, \(a = -0.05\) and \(b = 0.8\). Plugging these values into the formula gives:
\[
x = -\frac{0.8}{2 \cdot -0.05} = -\frac{0.8}{-0.1} = 8
\]
Thus, the ball reaches its maximum height at a distance of \(8\) feet from the player.