# 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?

Mar 11, 2018

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

$y = - \frac{16}{100} {x}^{2} + \frac{16}{5} x$

#### 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: $a {x}^{2} + b x + c$

Product of roots = $\frac{c}{a}$ = 0 therefore $c = 0$
Sum of roots = $- \frac{b}{a} = 20$ therefore $20 a + b = 0$

We are given a third equation from the maximum point:
When x=10, y=16, i.e. $16 = a \cdot {10}^{2} + b \cdot 10 + c$

Since $c = 0$, and as above:

$10 a + b = \frac{16}{10}$
$20 a + b = 0$

by subtraction: $- 10 a = \frac{16}{10}$
$a = - \frac{16}{100}$
therefore: $b = \frac{16}{5}$

Returning to our general form of the quadratic equation: $y = a {x}^{2} + b x + c$ we can sub in values for a and b to find the equation to be:

$y = - \frac{16}{100} {x}^{2} + \frac{16}{5} x$