Question

the equation for a projectile height versus time is h(t)= -16^2+ Vot+ho

a tennis ball machine serves a ball vertically into the air from a height of two feet, with an initial speed of 120 feet per second . after how many seconds does the ball attain its maximum height

Answer 1

$$h(t)=2+120t-16t^2, t\ge0$$

Find the first derivative with respect to $t$

$$h'(t)=120-32t$$

Find the critical number(s)

$$h'(t)=0=>120-32t=0$$

$$t=3.75\ s$$

Critical number: 3.75

If $0\le t<3.75, h'(t)>0, h(t)$ increases.

If $t>3.75, h'(t)<0, h(t)$ decreases.

The function $h(t)$ has a local maximum at $t=3.75.$ Since the function $h(t)$ has the only extremum for $t\ge0,$ then the function $h(t)$ has the absolute maximum at $t=3.75.$

The ball attains its maximum height after $3.75$ seconds .