Question

a boat is heading towards a lighthouse, where aaden is watching from a vertical distance of 103 feet above the water. aaden measures an angle of depression to the boat at point a to be 18°. at some later time, aaden takes another measurement and finds the angle of depression to the boat (now at point b) to be 75°. find the distance from point a to point b. round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Understand the right triangles

We have two right triangles here, both with the vertical side (height of the lighthouse) equal to 103 feet. The angle of depression is equal to the angle of elevation from the boat to Aaden's position (alternate interior angles). Let's denote the horizontal distance from the lighthouse to point A as \(x_A\) and to point B as \(x_B\).

Step2: Use tangent function for point A

For the angle of elevation \(18^\circ\) (same as angle of depression), \(\tan(18^\circ)=\frac{103}{x_A}\). Solving for \(x_A\), we get \(x_A = \frac{103}{\tan(18^\circ)}\).

Step3: Use tangent function for point B

For the angle of elevation \(75^\circ\) (same as angle of depression), \(\tan(75^\circ)=\frac{103}{x_B}\). Solving for \(x_B\), we get \(x_B=\frac{103}{\tan(75^\circ)}\).

Step4: Calculate the distance between A and B

The distance from A to B is \(x_A - x_B\) (since the boat is moving towards the lighthouse, \(x_A>x_B\)).

First, calculate \(\tan(18^\circ)\approx0.3249\) and \(\tan(75^\circ)\approx3.7321\).

Then, \(x_A=\frac{103}{0.3249}\approx317.0\) (rounded to one decimal place) and \(x_B = \frac{103}{3.7321}\approx27.6\) (rounded to one decimal place).

Now, the distance \(AB=x_A - x_B=317.0 - 27.6 = 289.4\)

Answer:

The distance from point A to point B is \(\boxed{289.4}\) feet.