2. Maximum mark: Sam and Markus are on holiday exploring the coast at Wollongong. Sam has climbed the 25 m high lighthouse, which stands at the top of a cliff. Markus is in a boat some distance offshore. He measures the angle of elevation to the base of the lighthouse is 6.4°, and the angle of elevation to Sam is 10.3 a Find the distance from Markus to the base of the cliff below the lighthouse. b Find the height of the cliff.

Answer 1

To solve this problem, we can use trigonometry. Let's call the distance from Markus to the base of the cliff "x" and the height of the cliff "h".

a) To find the distance from Markus to the base of the cliff, we can use the tangent function. The tangent of the angle of elevation to the base of the lighthouse is equal to the opposite side (h) divided by the adjacent side (x). Therefore:
tan(6.4°) = h/x

To find x, we rearrange the equation:
x = h / tan(6.4°)

b) To find the height of the cliff, we can use the tangent function again. The tangent of the angle of elevation to Sam is equal to the opposite side (h) divided by the adjacent side (x + 25) since Sam is standing on top of the lighthouse. Therefore:
tan(10.3°) = h / (x + 25)

To find h, we rearrange the equation:
h = tan(10.3°) * (x + 25)

Now we can substitute the value of x from equation a) into equation b) to find the height of the cliff.