The quadratic function h(t)=-16t^2 + 128t +144 models the ball's height about the ground, h(t), in feet, t, seconds after it was thrown.

Question

A person standing close to the edge on top of a 144-foot building throws a ball vertically upward. The quadratic function h(t)=-16t^2 + 128t +144 models the ball's height about the ground, h(t), in feet, t, seconds after it was thrown.

1. What is the maximum height of the ball?
2. How many seconds does it take until the ball hits the ground?

Answer 1

For part 1, imagine a downward facing parabola. The maximum height of the ball is at the top of the parabola. In other words, it is at the parabola's vertex. To find the vertex of a parabola, we can use the following formula,

( (-b/2a) , h(-b/2a) )

Note that b = 128 and a = -16 since the equation is given to us in standard form.

-b/2a = -128/(2)(-16) = 128/32 = 4

h(4) = 16(4)^2 + 128(4) +144 = 16(16) + 128(4) + 144 = 912

Therefore the maximum height is 912 feet,

For part 2, solve the equation for when the height function h(t) is 0,

h(t)=-16t^2 + 128t +144

-16t^2 + 128t +144 = 0

t^2 - 8t - 9 = 0 [divide by -16]

(t - 9)(t + 1) = 0

So t = 9 or t = -1.

Since time cannot be a negative value, we know the answer must by 9.

Therefore the ball reaches the ground at 9 seconds.