How to write the inequality of a parabola + more Algebra 1

Question

An engineer uses the equation below to model the height of a cannonball shot straight up from ground level. In the model the height y (in feet) is a function of x, the number of seconds after the cannonball is shot.

y = 96x - 16x²

Fill in the blanks below to write an inequality for all the values of x for which the cannonball is gaining height (going up). Then choose the best description of this inequality.

Inequality: ___ < x < ___

Description: ____

A: The function is increasing over these values of x

B: The function is decreasing over these values of x

Please fill in the blanks, thank you!

Answer 1

y=96x-16x^2

y'= 96-32x=0

x = 96/32= 3

0<x<3 are 3 seconds when height increases

decreasing when 3<x<6

but this is an algebra problem, not calculus

so solve by finding the x coordinate of the parabola's vertex, by completing the square

y = 96x -16x^2 = -16(x^2 -6x + 9) + 9(16) = -16(x-3)^2 + 144

vertex = (3,144) reaching 144 feet in 3 seconds

y = 96(3)-16(3^2) = 288 - 144 = 144

it hits the ground in 6 seconds. y=96(6)-16(6^2) = 0

it started at the ground at time x=0, y=96(0) -16(0^2) = 0

maximum height in 3 seconds when x=3, y =96(3)-16(3^2) = 144

increasing in height from x=0 to x=3

decreasing from x=3 to x=6