How Does Calculus Explain Uneven Water Distribution in Lawn Sprinklers?

[brewAP2010]

"Calculus of lawn sprinklers"

Homework Statement

A lawn sprinkler is constructed in such a way that dθ/dt is constant, where θ ranges between 45⁰ and 135⁰. The distance the water travels horizontally is x= (v^2sin2θ)/32, 45⁰ < θ < 135⁰ where v is the speed of the water. Find dx/dt and explain why this lawn sprinkler does not water evenly. What part of the lawn receives the most water?

Homework Equations

The Attempt at a Solution

If I’m not mistaken the velocity of the water should be a constant so v^2/32 is a coefficient, and when you derive dx/dt=(v^2/32)cos2θ(2dθ/dt).

 

[harp AP 2010]

You correctly derived the equation. Since the derivative is not constant there is an uneven distribution of water. To find where the most water goes you would need to integrate, or find where the change in x is smallest. This is at the endpoints, nearly 45 and 135, here is where the function is at its highest (greatest integral value) and the derivative is nearly zero.