Question

What is the maximum height, in feet, the ball will attain?

Answer 1

`h = 227\ ft`

Explanation:

We know that the equation that models the height of a projectile as a function of time is:

`h(t) = -16t ^ 2 +v_0t +h_0`

Where:

`v_0` is the initial velocity

`h_0` is the initial height of the projectile.

In our case, the height of the machine is 2 ft.

Then
`h_0 = 2\ ft`

The initial speed is 120 ft/s.

So the equation of the height for this case is:

`h(t) = -16t ^ 2 + 120t + 2`

This is a quadratic equation whose main coefficient is negative.

The maximum value of the function is at its vertex.

For a quadratic function of the form:

`at ^ 2 + bt + c`

the vertex of the equation is given by the expression:

`x =(-b)/(2a)`

`y = f((-b)/(2a))`

In this case:

`a = -16\\b = 120\\c = 2`

Then the maximum point occurs instantly:

`t = -(120)/(2(-16))\\\\t = 3.75\ s`

Finally the maximum atura is:

`h(3.75) = -16(3.75) ^ 2 +120(3.75) + 2`

`h = 227\ ft`