# A ball is thrown into the air with an upword velocity of 36 ft/s. Its hight H in feet after t seconds is given by the function h=-16t^2+36t+9. In how many seconds does the ball reach its maximum height? Round to the nearest hundredth if necessary. What is the balls maximum height

## To find the time at which the ball reaches its maximum height, we need to determine the time when the velocity of the ball becomes zero. This is because at the maximum height, the ball momentarily stops moving upwards before it starts descending.

Given that the ball's height is given by the function h = -16t^2 + 36t + 9, we can find the time at which the velocity is zero by taking the derivative of the height function with respect to time (t).

Differentiating the height function, we get:

h' = -32t + 36.

To find when the velocity is zero, we set h' = 0:

-32t + 36 = 0.

Solving for t:

-32t = -36,
t = -36 / -32,
t = 1.125 (rounded to the nearest hundredth).

Therefore, the ball reaches its maximum height after approximately 1.13 seconds.

To find the maximum height, we substitute this time back into the height function:

h = -16(1.13)^2 + 36(1.13) + 9.

Evaluating this expression:

h ≈ -16(1.2769) + 40.68 + 9,
h ≈ -20.4304 + 40.68 + 9,
h ≈ 29.2496 (rounded to the nearest hundredth).

Therefore, the ball reaches a maximum height of approximately 29.25 feet.

## To find the time at which the ball reaches its maximum height, we need to determine the vertex of the parabolic function h(t) = -16t^2 + 36t + 9.

The vertex of a parabola in the form y = ax^2 + bx + c is given by the x-coordinate:
x = -b / (2a)

In this case, a = -16 and b = 36. Plugging in these values, we have:
x = -36 / (2*(-16))
x = -36 / (-32)
x = 1.125

Therefore, it takes approximately 1.125 seconds for the ball to reach its maximum height.

To find the maximum height of the ball, we substitute the time value (1.125) into the function h(t):
h(1.125) = -16(1.125)^2 + 36(1.125) + 9

Calculating this expression:
h(1.125) = -16(1.265625) + 40.5 + 9
h(1.125) = -20.25 + 40.5 + 9
h(1.125) = 29.25

Therefore, the ball reaches its maximum height of 29.25 feet.

## You want the vertex of this parabola.

the x of the vertex is -b/2a
= -36/-32 = 1.125 seconds

plug that in
h = - 16(1.125)^2 + 36(1.125) + 9 = ....