Question

a boat is heading towards a lighthouse, whose beacon - light is 119 feet above the water. from point a, the boat’s crew measures the angle of elevation to the beacon, 5°, before they draw closer. they measure the angle of elevation a second time from point b at some later time to be 18°. find the distance from point a to point b. round your answer to the nearest foot if necessary.

Explanation:

Step1: Find the distance from the base of the lighthouse to point A

Let the distance from the base of the lighthouse to point A be \(x\).
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
For the angle of elevation of \(5^{\circ}\), \(\tan5^{\circ}=\frac{119}{x}\), so \(x = \frac{119}{\tan5^{\circ}}\).
Using a calculator, \(\tan5^{\circ}\approx0.0875\), then \(x=\frac{119}{0.0875}=1360\) feet.

Step2: Find the distance from the base of the lighthouse to point B

Let the distance from the base of the lighthouse to point B be \(y\).
For the angle of elevation of \(18^{\circ}\), \(\tan18^{\circ}=\frac{119}{y}\), so \(y=\frac{119}{\tan18^{\circ}}\).
Using a calculator, \(\tan18^{\circ}\approx0.3249\), then \(y = \frac{119}{0.3249}\approx366.3\) feet.

Step3: Calculate the distance from point A to point B

The distance from point A to point B is \(d=x - y\).
Substitute \(x = 1360\) and \(y\approx366.3\) into the formula, \(d=1360- 366.3=993.7\approx994\) feet.

Answer:

The distance from point A to point B is \(994\) feet.