Question

a boat is heading towards a lighthouse, where benjamin is watching from a vertical distance of 113 feet above the water. benjamin measures an angle of depression to the boat at point a to be 17°. at some later time, benjamin takes another measurement and finds the angle of depression to the boat (now at point b) to be 70°. 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 (\(L\)) to point \(A\) be \(x\).
We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). The angle of depression to point \(A\) is \(17^{\circ}\), and the vertical distance (opposite side) is \(113\) feet.
\(\tan(17^{\circ})=\frac{113}{x}\), so \(x = \frac{113}{\tan(17^{\circ})}\).
Using a calculator, \(\tan(17^{\circ})\approx0.3057\), then \(x=\frac{113}{0.3057}\approx370\) feet.

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

Let the distance from the base of the lighthouse (\(L\)) to point \(B\) be \(y\).
The angle of depression to point \(B\) is \(70^{\circ}\), and the vertical distance (opposite side) is still \(113\) feet.
\(\tan(70^{\circ})=\frac{113}{y}\), so \(y=\frac{113}{\tan(70^{\circ})}\).
Using a calculator, \(\tan(70^{\circ})\approx2.7475\), then \(y = \frac{113}{2.7475}\approx41\) feet.

Step3: Calculate the distance from \(A\) to \(B\)

The distance from \(A\) to \(B\) is \(d=x - y\).
Substitute \(x = 370\) and \(y = 41\) into the formula: \(d=370-41 = 329\) feet.

Answer:

\(329\) feet