I never learned this skill

Question

On one of the banks of the Chicago River, there is a water cannon, called the Water Arc, that sprays recirculated water across the river. The path of the Water Arc is given by the model = -0.006x² + 1.2x + 10. Where x is the distance in feet across the river, and y is the height of the arc in feet. What is the maximum height of the water sprayed from the Water Arc?

Answer 1

Y = -0.006x² + 1.2x + 10.

let a=-0.006, b=1.2 and c=10

Use the quadratic formula to find 2 values for x that result in y=0

[-b +/- sqrt(b² - 4ac)] / 2a

Plug in values of a, b, c to get

[-1.2 +/- sqrt(1.2² - 4(-0.006)(10)]/2(-0.006)

= [-1.2 +/- sqrt(1.44 + 0.24)] / -0.012

= [-1.2 +/- sqrt(1.68)] / -0.012

[-1.2 - sqrt(1.68)] / -0.012 = 208.0123 feet

[-1.2 + sqrt(1.68)] / -0.012 = -8.0123 feet

The distance the water travels has to be x=208.0123 feet as the other option, x= -8.0123, is impossible.

Because the equation is quadratic, it is like a parabola in that the distance x required to reach a maximum (or minimum) is 1/2 the sum between the 2 values of x where y=0.

208.0123 + -8.0123 = 200. 1/2 this sum is 100 feet

When x=100 feet, y reaches a maximum height of -0.006(100²) + 1.2(100) + 10 feet.

Y = 70 feet.