Question

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Part 1
A toy rocket is shot vertically into the air from a launching pad 4 feet above the ground with an initial velocity of 152 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+152t+4. How long will it take the rocket to reach its maximum​ height? What is the maximum​ height?

Answer 1

To find the maximum height of the rocket, we first need to determine the time at which the rocket reaches its maximum height. We can do this by finding the value of t when the derivative of the height function, h(t), is equal to zero.

The derivative of h(t) = -16t^2 + 152t + 4 is given by:

h'(t) = -32t + 152

Setting h'(t) equal to zero:

-32t + 152 = 0

Solving for t, we get:

-32t = -152
t = 4.75

So, it will take the rocket 4.75 seconds to reach its maximum height.

To find the maximum height, we substitute the value of t into the height function:

h(4.75) = -16(4.75)^2 + 152(4.75) + 4
h(4.75) = -16(22.5625) + 722 + 4
h(4.75) = -361 + 722 + 4
h(4.75) = 365

Therefore, the maximum height of the rocket is 365 feet.