Question

Derek kicks a soccer ball from the ground. The height of the ball can be modeled by the equation h(t)=−8t^2+32t, where h represents the height (in feet) of the ball from the ground and t represents the time (in seconds) after Derek kicks the ball. What is the maximum height of the ball and when does it occur?

Answer 1

The maximum height reached by the ball is 32 feet, and it occurs 2 seconds after it is kicked. This is found by analyzing the vertex of the parabola represented by the equation h(t).

Step-by-step explanation:

The maximum height of the soccer ball and the time at which it occurs can be determined by analyzing the vertex of the parabola represented by the equation h(t) = -8t^2 + 32t. The equation is in the form of at^2 + bt + c, where the vertex of the parabola occurs at t = -b/(2a). In this case, a = -8 and b = 32, leading to the vertex at t = -32/(2(-8)) = 2 seconds. The maximum height can be found by substituting the time back into the equation: h(2) = -8(2)^2 + 32(2) = -32 + 64 = 32 feet. Therefore, the maximum height the ball reaches is 32 feet, and it occurs at 2 seconds after Derek kicks the ball.