Question

A toy rocket is shot vertically into the air from a launching pad 8 feet above the ground with an initial velocity of 128 feet per second. The height​ h, in​ feet, of the rocket above the ground at t seconds after launch is given by the function h equals negative 16 t squared plus 128 t plus 8h=−16t2+128t+8. How long will it take the rocket to reach its maximum​ height? What is the maximum​ height?

Answer 1

The rocket will be reach its maximum​ height at t= 4 seconds

The maximum height 'h' = 264 feet

Explanation:

Step(i):-

Given the function 'h' = -16 t² + 128 t +8 ...(i)

Differentiating equation (i) with respective to 't' we get

`(d h)/(d t) = - 16 ( 2 t) + 128(1)` ...(ii)

Now equating Zero , we get

`(d h)/(d t) = -32 t + 128 =0`

- 32 t =-128

t = 4

Step(ii):-

Again Differentiating equation (ii) with respective to 't' we get

`(d^2 h)/(d t^2) = -32 (1) <0`

The maximum height at t= 4

The rocket will be reach its maximum​ height at t= 4 seconds

The maximum height

h = -16 (4)² + 128 (4) +8

= 264

Conclusion:-

The rocket will be reach its maximum​ height at t= 4 seconds

The maximum height 'h' = 264