Question

3. A ball was thrown into the air with an initial velocity of 72 feet per second. The height of the

ball after t seconds is represented by the equation: h= -16t2 + 72t. The graph of the function
is provided.
(a) What is the maximum height the ball reaches?
(b) When does the ball reach its maximum height?
(c) Describe the domain for this problem:
(d) Describe the range for this problem:
(e) How long is the ball in the air?

Answer 1

A. The maximum heigh is 81 feet

B. The ball will reach the maximum heiight at 2.25 seconds

C. The domain of the function is
`t\in [0,4.5]`

D. The range of the function is
`h(t)\in [0,81]`

E. 4.5 seconds

Explanation:

A ball was thrown into the air with an initial velocity of 72 feet per second. The height of the ball after t seconds is represented by the equation:

`h= -16t^2 + 72t`

The maximum height will be at parabola's vertex. Find it:

`t_v=(-b)/(2a)\\ \\t_v=-(72)/(2\cdot (-16))=(72)/(32)=(9)/(4)=2.25\\ \\h(t_v)=-16\cdot t_v^2+72\cdot t_v\\ \\h(2.25)=-16\cdot 2.25^2+72\cdot 2.25=81`

A. The maximum heigh is 81 feet

B. The ball will reach the maximum heiight at 2.25 seconds

C. Find where the parabola intersects the x-axis:

`h=0\Rightarrow -16t^2+72t=0\\ \\t(-16t+72)=0\\ \\t_1=0\ \text{or}\ 16t=72,\ t_2=(9)/(2)=4.5`

So, the domain of the function is
`t\in [0,4.5]`

D. The range of the function is

`h(t)\in [0,81]`

E. The ball is at the air for 4.5 seconds