A toy rocket launched straight up from the ground with an initial velocity of 80 ft/s returns to the ground after 5 s. The height of the rocket t seconds after launch is modeled by the function f(t)=−16t^2+80t . What is the maximum height of the rocket, in feet? Enter your answer in the box.

The height of the rocket is modeled by the function:

[tex]f(t)=-16 t^{2}+80t [/tex]

If we observe this equation, we see that the function is quadratic. The shape of the quadratic function is parabolic and the maximum or minimum value of a parabola always lies at its vertex. In the given function, since the co-efficient of leading term (t²) is negative, so this parabola will have a maximum value at its vertex. 

The vertex of parabola is given by: 

[tex]( \frac{-b}{2a}, f( \frac{-b}{2a})) [/tex]

b is the coefficient of t term. So b = 80
a is the coefficient of squared term. So a= - 16

So,

[tex] \frac{-b}{2a}= \frac{-80}{2(-16)}= \frac{5}{2}=2.5 [/tex]

This means at 2.5 sec the height of rocket will be maximum. The maximum height will be:

[tex]f(2.5)=-16 (2.5)^{2}+80(2.5)=100 [/tex]

Therefore, the maximum height of the rocket will be 100 feet.