Jim held a firehose whose spray formed a parabola that spanned 20m. The maximum height of the spray is 16m. What is the quadratic equation that models the path of the spray?

1 Answer
Mar 11, 2018

Answer:

graph{-0.16x^2+3.2x [-4.41, 27.63, 1.96, 17.98]}

#y=-16/100x^2+16/5x#

Explanation:

Assuming Jim is standing at the point (0,0) facing to the right, we are told that the two intercepts (roots) of the parabola are at (0,0) and (20,0). Since a parabola is symmetrical, we can infer that the maximum point is in the middle of the parabola at (10,16).

Using the general form of the parabola: #ax^2+bx+c#

Product of roots = #c/a# = 0 therefore #c=0#
Sum of roots = #-b/a=20# therefore #20a+b=0#

We are given a third equation from the maximum point:
When x=10, y=16, i.e. #16=a*10^2+b*10+c#

Since #c=0#, and as above:

#10a+b=16/10#
#20a+b=0#

by subtraction: #-10a=16/10#
#a=-16/100#
therefore: #b=16/5#

Returning to our general form of the quadratic equation: #y=ax^2+bx+c# we can sub in values for a and b to find the equation to be:

#y=-16/100x^2+16/5x#