Question

a boat is heading towards a lighthouse, where alexa is watching from a vertical distance of 125 feet above the water. alexa measures an angle of depression to the boat at point a to be 8°. at some later time, alexa 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 \(A\) to \(L\)

We know that the angle of depression from the lighthouse to point \(A\) is \(8^{\circ}\). The vertical distance (height of the lighthouse) \(h = 125\) feet.
Since the angle of depression is equal to the angle of elevation from the boat to the top of the lighthouse. Using the tangent function \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), for \(\theta = 8^{\circ}\) and opposite side \(h = 125\) feet. Let the distance from \(A\) to \(L\) be \(x\). Then \(\tan(8^{\circ})=\frac{125}{x}\), so \(x=\frac{125}{\tan(8^{\circ})}\).
Using a calculator, \(\tan(8^{\circ})\approx0.1405\), then \(x=\frac{125}{0.1405}\approx889.7\) feet.

Step2: Find the distance from \(B\) to \(L\)

For the angle of depression \(70^{\circ}\) (angle of elevation from \(B\) to the top of the lighthouse). Let the distance from \(B\) to \(L\) be \(y\). Using the tangent function \(\tan(70^{\circ})=\frac{125}{y}\), so \(y = \frac{125}{\tan(70^{\circ})}\).
Using a calculator, \(\tan(70^{\circ})\approx2.747\), then \(y=\frac{125}{2.747}\approx45.5\) feet.

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

The distance from \(A\) to \(B\) is \(d=x - y\).
Substitute \(x\approx889.7\) and \(y\approx45.5\) into the formula: \(d=889.7- 45.5=844.2\approx844\) feet.

Answer:

\(844\) feet