Question

2. The formula h(t) = -16t² + 32t + 80 gives the height b above the ground, in feet (ft), of an object thrown, at t = 0, straight upward from the top of an 80 ft building. What is the highest point reached by the object? How long does it take the object to reach its highest point?

Answer 1

Find the first derivative with respect to $t$

$$h'(t)=-32t+32, t\geq 0$$

Find the critical number(s)

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

$$t=1$$

If $0<t<1, h'(t)>0, h(t)$ increases.

If $t>1, h'(t)<0, h(t)$ decreases.

The function $h(t)$ has a local maximum at $t=1.$

Since the function $h$ has the only extremum, the function $h(t)$ has the absolute maximum at $t=1.$

i)

$$h(1)=-16(1)^2 + 32(1) + 80=96(ft)$$

The highest point reached by the object is 96 ft above the ground.

ii) It takes the object 1 sec to reach its highest point.