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The function h(t) = -16t^2 + 30t + 24 is a quadratic function, whose plot is a parabola opened down.
This quadratic function has the maximum at the value of its argument t = ,
where "a" is the coefficient at t^2 and "b" is the coefficient at t.
In your case, a= -16, b= 30, so the function gets the maximum at t = = 0.9375.
So, the ball gets the maximum height 0.9375 seconds after is hit straight up.
The maximum height then is h(0.9375) = -16*0.9375^2 + 30*0.9375 + 24 = 38.0625 ft.
ANSWER
Solved.
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On finding the maximum/minimum of a quadratic function, see 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.
On solving similar problems to yours in this post, see the lessons
- Problem on a projectile moving vertically up and down
- Problem on an arrow shot vertically upward
- Problem on a ball thrown vertically up from the top of a tower
- Problem on a toy rocket launched vertically up from a tall platform
in this site.