An apple is launched directly upward at 64 feet per second from an 80-foot tall platform. The equation for this apple’s height h at time t seconds after launch h = -16^2+ 64t+ 80.

Question

a. Find the maximum height of the apple.

b. How many seconds will it take for the apple to reach the ground?

Answer 1

a. To find the maximum height of the apple, we need to find the vertex of the parabolic equation h = -16t^2 + 64t + 80.

The vertex of a parabolic equation in the form h = at^2 + bt + c is given by the formula t = -b / (2a).

In this case, a = -16 and b = 64. Plugging these values into the formula, we get:

t = -64 / (2*(-16))
t = -64 / -32
t = 2

Now, we can calculate the maximum height by plugging t = 2 into the equation:

h = -16(2)^2 + 64(2) + 80
h = -16(4) + 128 + 80
h = -64 + 128 + 80
h = 144

Therefore, the maximum height of the apple is 144 feet.

b. To find out how many seconds it will take for the apple to reach the ground, we need to solve for when h = 0 in the equation h = -16t^2 + 64t + 80.

-16t^2 + 64t + 80 = 0
Dividing by -16 to simplify the equation:
t^2 - 4t - 5 = 0
This is now a quadratic equation that can be factored:
(t - 5)(t + 1) = 0
Setting each factor to zero:
t - 5 = 0 or t + 1 = 0
t = 5 or t = -1

Since time cannot be negative in this case, the apple will reach the ground after 5 seconds.

Therefore, it will take 5 seconds for the apple to reach the ground.