Question

A baseball is dropped from a glider 500 feet above the ground. The height (in feet) above the ground is modeled by the function ℎ() = −162 + 500, where is the time in seconds. A) What is the height of the ball after 2 seconds? B) How long will it take for the ball to fall to a height of 100 feet above the ground? C) How long will it take the ball to reach the ground?

Answer 1

A) 436 ft

B) t = 5 sec

C) 5.60 s

Step-by-step explanation:

A)

The height of the ball at time t is given by the equation

$h(t)=-16t^2+500$

where

$-16 ft/s^2$ is the acceleration of the ball (acceleration of gravity, downward)

+500 is the initial height of the ball, at time t = 0

Here we want to find the height of the ball after 2 seconds, so at a time of

t = 2 s

Substituting into the equation, we find:

$h(2)=-16\cdot 2^2+500=436 ft$

B)

Here we want to find the time it takes for the ball to fall to a height of 100 feet above the ground, so the time t at which

h(t) = 100 ft

As stated in the text of the question, the height of the ball at time t is given by

$h(t)=-16t^2+500$

Since

h(t) = 100, we have

$100=-16t^2+500$

And solving for t we find:

$16t^2=400\\t^2=25\\t=\pm 5$

So, the correct solution is the positive one:

t = 5 sec

C)

The ball reaches the ground when the height of the ball has became zero:

$h(t) = 0$

The height of the ball at time t is given by

$h(t)=-16t^2+500$

And substituting

h(t) = 0

We get

$0=-16t^2+500$

And solving the equation for t, we find the time t at which the ball reaches the ground:

$16t^2=500\\t^2=31.25\\t=\pm 5.6 s$

So, the correct solution is the positive one:

t = 5.60 s