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.