I am currently doing a math exploration and I need help in determining how to find the angle of a spigot that would maximize the area under a parabolic fountain. I thought of this topic to investigate, but I have no idea how to start! I know that this somehow involves integration.
I have trouble trying to find the angle and relating that to the water being ejected from the spigot.
Presumably, the spigot is at ground height, so that the parabolic trajectory of the water is given by the parameterization $$ x(t) = (v_0\cos \theta )\\ y(t) = (v_0 \sin \theta)t -gt^2 $$ Solving for $t$ in terms of $x$ gives $$ t = \frac{x}{v_0 \cos\theta} \implies\\ y(x) = -\left(\frac{g}{v_0^2 \cos^2\theta}\right)x^2 + (\tan \theta) x $$ Now, find the value of $\theta$ that maximizes the area beneath this curve.