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It is about finding the vertex (the maximum) of the quadratic function
s(t) = -16*t^2 + 4t.
In the given case you can present the function as the product
s(t) = -4t*(4t-1)
of two factors -4t and (4t-1). Then it is clear that the quadratic function has the roots at t = 0 and t = 1/4.
Then the midpoint t = 1/8 is the point where the quadratic function reaches its maximum.
So the time to get maximum height is 1/8 of a second.
Then the maximum height is = = .
Answer. The maximum height is of the foot and it reaches at t = 1/8 of a second.
Plot h(t) =
There is another way to analyze the problem.
It is described in lessons
- Problem on an arrow shot vertically upward
- Problem on a ball thrown vertically up from the top of a tower
in this site.
In any case, you need to know how to find the maximum/minimum of a quadratic function presented in the general form.
You will find it in the lessons
- HOW TO complete the square to find the minimum/maximum of a quadratic function
- Briefly on finding the minimum/maximum of a quadratic function
- HOW TO complete the square to find the vertex of a parabola
- Briefly on finding the vertex of a parabola
in this site.