The maximum height of the water balloon : 450 m
Quadratic function is a function that has the term x²
The quadratic function forms a parabolic curve
The general formula is
f (x) = ax² + bx + c
where a, b, and c are real numbers and a ≠ 0.
The parabolic curve can be opened up or down determined from the value of a. If a is positive, the parabolic curve opens up and has a minimum value. If a is negative, the parabolic curve opens down and has a maximum value
So the maximum is if a <0 and the minimum if a> 0.
The formula for finding the coordinates of the maximum and minimum points of the quadratic function is the same.
The maximum / minimum point of the quadratic function is
[tex]\rm -\dfrac{b}{2a},-\dfrac{D}{4a}[/tex]
Where
D = b²-4ac
The function h (t) = -16x²+160x+50
so the value of a <0, then it has a maximum value
Because we are looking for maximum height, then we find the value of the y coordinate, with the formula
[tex]\rm -\dfrac{D}{4a}[/tex]
Questions we might add :
What was the maximum height of the water balloon after it was thrown?
We can also use the first derivative of the above function to find the maximum value
[tex]\tt h'=0=-32x+160\\\\-32x=-160\\\\x=5[/tex]
input to function :
[tex]\tt h=-16x^2+160x+50\\\\h=-16(5^2)+160(5)+50\\\\h=-400+800+50=\boxed{\bold{450~m}}[/tex]
