Question

a boat is heading towards a lighthouse, where michael is watching from a vertical distance of 139 feet above the water. michael measures an angle of depression to the boat at point a to be 17°. at some later time, michael 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 horizontal distance to point A

Angle of depression = angle of elevation from boat to Michael, so use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. For point A:
$\tan(17^\circ)=\frac{139}{d_A}$
$d_A=\frac{139}{\tan(17^\circ)}$
$d_A\approx\frac{139}{0.3057}\approx454.7$ feet

Step2: Find horizontal distance to point B

For point B, angle of depression is $70^\circ$:
$\tan(70^\circ)=\frac{139}{d_B}$
$d_B=\frac{139}{\tan(70^\circ)}$
$d_B\approx\frac{139}{2.7475}\approx50.6$ feet

Step3: Calculate distance AB

Subtract the two horizontal distances:
$AB = d_A - d_B$
$AB\approx454.7 - 50.6 = 404.1$

Answer:

404 feet