Answer
Maximum height reached by the rocket = 444 ft.
Step-by-step explanation
The height of the rocket, y, at any time x is given as
y = -16x² + 152x + 83
We are now asked to find the maximum height reached by the rocket.
For any function, the maximum value (height in this case) occurs at the point where the first derivative of the function is 0 and the second derivative of the function is negative.
That is,
(dy/dx) = 0
(d²y/dx²) < 0
y = -16x² + 152x + 83
(dy/dx) = -32x + 152
At maximum height, (dy/dx) = 0
-32x + 152 = 0
-32x = -152
Divide both sides by -32
(-32x/-32) = (-152/-32)
x = 4.75 seconds
So, this gives the time when the maximum height occurs.
To find this exact height, we will put this term for x in the original equation and solve for y when x = 4.75 s
y = -16x² + 152x + 83
y = -16 (4.75)² + 152 (4.75) + 83
y = -16 (22.5625) + 722 + 83
y = -361 + 722 + 83
y = 444.0 ft.
To confirm that this is indeed the maximum height reached by the rocket, we will compute the second derivative of the function.
y = -16x² + 152x + 83
(dy/dx) = -32x + 152
(d²y/dx²) = -32 < 0
This is indeed the maximum value of the given height function.
Hope this Helps!!!