Sketch a diagram describing the given situation. Then recall the definition of tangent.
Plum Island Lighthouse.
Practice makes perfect
We are given that Little Gull Island Lighthouse shines a light from a height of 91 feet with a 6^(∘) angle of depression and that the Plum Island Lighthouse, which is located 1800 feet from the first one, shines a light from a height of 34 feet with a 2^(∘) angle of depression. Let's sketch a diagram describing this situation.
We are asked to determine which of the lighthouses shines a light that reach a boat that sits exactly between them. Let x and y represent the range of lights of Little Gull Island and Plum Island.
To find the values of x and y, we can use one of the trigonometric ratios. Let's recall that the tangent of ∠A is the ratio of the leg opposite ∠A to the leg adjacent ∠A. Using this definition, we can create equations for tan 6^(∘) and tan 2^(∘).
tan 6^(∘)=91/x
tan 2^(∘)=34/y
Let's solve the above equations. We will start with the first one.
We know that the boat is exactly between the lighthouses, so it lies 18002=900 feet from each of the lighthouses. Let's compare this distance to the range of light of each lightouse.
865.8 < 900
973.6>900
Since Plum Island Lighthouse's light has a range that is greater than 900, it will reach the boat. However, as the Little Gull Island's light has a range that is less than the distance to the boat, the light will not reach a boat.
