Question

a yacht is anchored 90 feet offshore from the base of a lighthouse. the angle of elevation from the boat to the top of the lighthouse is 26 degrees. the distance between the yacht and the top of the lighthouse is about 100 feet. which of these is nearest to the height of the lighthouse?

Explanation:

Step1: Use trigonometric function

We know that in a right - triangle (formed by the yacht, the base of the lighthouse and the top of the lighthouse), if the angle of elevation is \(\theta = 26^{\circ}\), the adjacent side \(x = 90\) feet (distance from yacht to base of lighthouse) and we want to find the opposite side \(y\) (height of the lighthouse). We can use the tangent function \(\tan\theta=\frac{y}{x}\).
So, \(y=x\tan\theta\).

Step2: Substitute the values

Substitute \(x = 90\) and \(\theta = 26^{\circ}\). We know that \(\tan(26^{\circ})\approx0.4877\).
Then \(y = 90\times\tan(26^{\circ})\).
\(y=90\times0.4877 = 43.893\approx45\)

Answer:

45 feet