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# Garden Sprinkler Mathematical Modelling (Trigonometric Substitutions)?

## There are various types of garden sprinkler systems on the market. In our analysis here we will consider a very basic design, a cylinder with holes made for water. The cylinder (of negligible size) distributes water to our rectangular lawn by rotating around its axis at a constant frequency. The cylinder rotates so that the maximal water angle is 45◦. The angle θ = 0 corresponds to a vertical shoot. We assume no air resistance in our model. It is clear that the lawn is not watered uniformly. The amount water that falls on the ground near vertical shoot is less than the amount of water falling down at the extremes. This is due to the fact that the water moves quicker on the ground near the vertical shoot. We will now analyse the water distribution on our rectangular lawn. (a) Let (v)0 be the (constant) speed of water departure from the cylinder holes and let g denote the acceleration due to gravity. Show that the water reach on the ground is given by (x)0 = [v(0)^2]/g (sin2θ) where the angle θ is the angle of the holes in the cylinder from the vertical. Thus the maximal water reach on the ground is given by x(0)=(v(0)^2]/g (b) Let f be the constant frequency of the cylinder motion, one cycle θ ∈ [−π/4, π/4]. Let k ∈ [0, 1] denote the fraction of the ground distance cover by water starting from zero (vertical) all the way to the maximal ground reach to x(0). Show that the time t(k) it takes to cover the ground from zero to kx(0) is given by: t(k) = [1/(4πf)]sin^-1(k) in particular, we have t(1)=(1/8f). Hint set x(0)= 1 and [v(0)^2]/g = 1 for this. (c) Let pk denote the percentage of total watering for the portion of our rectangular lawn corresponding to the values [−k, k]. Show that p(k) = (200/π)sin^-1(k) Hint: the amount of water in the region [−k, k] is proportional to t(k). (d) Show that the percentage of watering for the inner half of the lawn is about 33 % (one third). (e) Find the variable frequency f = f(θ) so that the lawn gets watered evenly.

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