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A toy rocket is shot vertically into the air from a launching pad 6 feet above the ground with an initial velocity of 88 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(t)=−16t2+88t+6. How long will it take the rocket to reach its maximum​ height? What is the maximum​ height?

The rocket reaches its maximum height at ? ​second(s) after launch.
(simplify your answer.)

Respuesta :

shubhamchouhanvrVT

a) The time taken by the rocket to reach the maximum height is 2.75 seconds.

b) The maximum height of the rocket will be 127 feet.

What is speed?

Speed is defined as the ratio of the time distance travelled by the body to the time taken by the body to cover the distance.

Given that:-

  • A toy rocket is shot vertically into the air from a launching pad 6 feet above the ground with an initial velocity of 88 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(t)=−16t²+88t+6.

a) The time taken by the rocket will be calculated as:-

We have a function h(t)=−16t²+88t+6. Now by taking the derivative of the function with respect to time and equating it to zero we will get the maximum time in which the rocket will reach the maximum height.

h(t)  =  -16t² + 88t + 6

h'(t)    =     0

-32t   +   88   =  0

32t      =      88

t     =    88 / 32    =  2.75 seconds

So the maximum time is 2.75 seconds.

b) Maximum height will be calculated as:-

Put the value of maximum time in the function h(t)  =  -16t² + 88t + 6 we will get the maximum height.

h(t)  =  -16t² + 88t + 6

h(2.75) = -16(2.75)² + ( 88 x 2.75 ) + 6

h(2.75) = 127 feet

Therefore the time taken by the rocket to reach the maximum height is 2.75 seconds and the maximum height of the rocket will be 127 feet.

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