Cool the Pool (A) ** Calculus Emphasis: Optimization. Skills Emphasized: Circle and triangle properties, trigonometry, and the derivative on a closed interval. Note: It is common for swimming pools in the southwest to have some sort of aerator device built into the deck or free-standing in order to spray a jet of water. This is often used to cool the pool when the water gets really warm due to high temperatures. One type of aerator jet sprays a fan pattern which can be adjusted to make the fan pattern wider or narrower. In essence, the angle at the jet can lie between 0 and 180 degrees. As the angle of the aerator is decreased, the spray extends further across the pool. The Problem. Suppose an adjustable aerator is located on the edge of a circular pool of radius 10 feet. The water pressure is such that when the angle is very small the jet of spray has length nearly equal to the diameter of the pool. As the angle is increased the spray forms a fan pattern which has decreasing radial length and the area of coverage by the spray can be considered the sector of a circle with center at the aerator. For what angle will the aerator produce a fan pattern that covers the most area of the pool? Pool Area of Coverage Aerator