Answer:
2344 feet is the max height
Step-by-step explanation:
1) we find the function.
the function will be a parabola with a y intercept of 40.
the max height will be at the vertex.
f(x) = ax^2 + bx + 40
2) we find the derivative
f'(x) = 2ax + b
3) since the velocity is 96 at x = 0, we get the function:
f'(x) = 2ax + 96
4) find the x intercept of the velocity function
0 = 2ax + 96
-96 = 2ax
-48/a = x
This is the time in seconds when the max height is reached.
5) we finish the equation f(x)
since the antiderivative of 96 is 96x, we can write the equation:
f(x) = -ax^2 + 96x + 40
a will be negative since the parabola is opening downwards.
6) Since -48/a and -b/2a are both equal, a = 1
f(x) = -x^2 + 96x + 40
7) substitute 48 into x
-48/a = 48
f(48) = -(48^2) + 96(48) + 40
f(48) = -2304 + 4608 + 40
f(48) = 2344
2344 feet is the max height
