At what angle of inclination should she set the sprinkler?

Question

Alayna is trying to determine the angle at which to aim her sprinkler nozzle to water the top of a 5 ft tree in her yard. Assuming the water takes a straight diagonal path and the sprinkler is on the ground 4 ft from the base of the tree, at what angle of inclination should she set the sprinkler?

Answer 1

Good afternoon and thank you for the question:

The answer is 51°.

Below is an explanation of how I arrived at the answer:

We know that the sprinkler is on the ground, the water will travel in a straight, diagonal path to the top of the tree, the sprinkler is 4 feet from the tree, and the tree is 5 feet high. Hence, we can assume the height of the tree, the distance between the tree base and the sprinkler, and the path of the water from the sprinkler to the top of the tree form a right triangle. The angle we desire, θ, is the angle between the ground and the path of the water from the sprinkler to the top of the tree. The side opposite this angle, θ, is the tree, measuring 5 feet. The side adjacent to the angle is the distance between the sprinkler and the base of the tree, measuring 4 feet. Hence,

tan(θ) = 5/4 because tangent is the ratio between the opposite side to an angle and the adjacent side to an angle. Finally, we use the inverse tangent to solve for, θ:

θ = tan^-1(5/4) ≈ 51^°.