Question

a water balloon was thrown from the window. The height of the water balloon over time can be modeled by the function y=-16square2+160x+50

Answer 1

The maximum height of the water balloon : 450 m

Further explanation  

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  

$\rm -\dfrac{b}{2a},-\dfrac{D}{4a}$

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  

$\rm -\dfrac{D}{4a}$

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

$\tt h'=0=-32x+160\\\\-32x=-160\\\\x=5$

input to function :

$\tt h=-16x^2+160x+50\\\\h=-16(5^2)+160(5)+50\\\\h=-400+800+50=\boxed{\bold{450~m}}$