Answer:
The ball reaches its maximum height at 2.5 seconds
The maximum height is 100 feet
Step-by-step explanation:
Maximum Value of Functions
We use the derivative of a function to find its maximum or minimum value over a given interval.
Given a function y=f(x), the first derivative criterion establishes if x=a is such that:
f'(a)=0, and f''(a) < 0 then x=a is a maximum of f.
The height h in feet of a ball after t seconds is:
[tex]h(t)=-16t^2+80t[/tex]
Find the first derivative:
[tex]h'(t)=-32t+80[/tex]
Equate to 0:
[tex]-32t+80=0[/tex]
Subtract 80:
[tex]-32t=-80[/tex]
Divide by -32:
[tex]t = -80 / (-32) = 2.5[/tex]
t=2.5 seconds
Find the second derivative:
h''(t)=-32
Since h''(t) is always negative, then
The ball reaches its maximum height at 2.5 seconds
To find the value of the maximum height, substitute t=2.5 into the function:
[tex]h(2.5)=-16(2.5)^2+80*2.5=-100+200 = 100[/tex]
The maximum height is 100 feet
