Question

1. the fisgard lighthouse near victoria, bc, was the first lighthouse built on canadas west coast. its role is to usher ships into the esquimalt harbour. (today, it is also an historical site.) suppose a ship was coming into the harbour and measured the angle of elevation to the light on the lighthouse as 4°. the light of the lighthouse is 21.6 m above sea level.

a. draw a diagram of the ship and the lighthouse with dimensions labelled.
b. determine the distance from the base of the lighthouse (the rocks) to the ship. round to the nearest metre.

Explanation:

Step1: Recall the tangent function

In a right - triangle, the tangent of an angle of elevation \(\theta\) is given by \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, the angle of elevation \(\theta = 4^{\circ}\), the height of the lighthouse (opposite side) \(h = 21.6\) m, and let the distance from the base of the lighthouse to the ship (adjacent side) be \(x\). So, \(\tan\theta=\frac{h}{x}\).

Step2: Solve for \(x\)

We know that \(\tan4^{\circ}=\frac{21.6}{x}\). Since \(\tan4^{\circ}\approx0.0699\), we can rewrite the equation as \(x=\frac{21.6}{0.0699}\).

Step3: Calculate \(x\)

\(x=\frac{21.6}{0.0699}\approx309\) m.

Answer:

The distance from the base of the lighthouse (the rocks) to the ship is approximately \(309\) m.