Question

Determine the maximum height a baseball reaches as it is hit if its height can be represented by a function s(t)= -5t^2 +55t +2?

Answer 1

We use the derivative to find `max(s) = 153.25`

Explanation:

The equation for the height of the baseball is a quadratic which should have a maximum where the baseball reaches.

graph{-5x^2 +55x +2 [-1, 15, -25, 199]}

What we want to know is the `y`-value at the top of the parabola, i.e. the point where the slope is zero. We can find the slope by differentiating the equation:

`s' =d/dt s(t)= d/dt( -5t^2 +55t +2) = -10t+55`

Solving for the time where the slope equals zero

`0=-10t+55 -> t = 5.5`

Substituting back into the equation for `s`

`s(5.5)= -5*(5.5)^2 +55*5.5 +2 = 153.25`

Answer 2

`153.25" unit"`.

Explanation:

Recall that, for `s_max, s'(t)=0, and, s''(t) lt 0`.

`s(t)=-5t^2+55t+2`.

`:. s'(t)=-10t+55, and, s''(t)=-10`.

`:. s'(t)=0 rArr -10t+55=0 rArr t=11/2`.

Since, `s''(t) lt 0, AA t in RR`, we find that,

`s_max=s(11/2)=-5(11/2)^2+55(11/2)+2`.

`rArr s_max=613/4=153.25" unit"`.