Question

a boat is heading towards a lighthouse, where feng is watching from a vertical distance of 127 feet above the water. feng measures an angle of depression to the boat at point $a$ to be $5^\circ$. at some later time, feng takes another measurement and finds the angle of depression to the boat (now at point $b$) to be $57^\circ$. find the distance from point $a$ to point $b$. round your answer to the nearest tenth of a foot if necessary.
answer attempt 1 out of 2
feet

Explanation:

Step1: Define variables for distances

Let $x$ = horizontal distance from lighthouse to point $A$, $y$ = horizontal distance from lighthouse to point $B$. The vertical height is $127$ ft, equal for both triangles.

Step2: Calculate distance to point A

Use tangent of angle of depression (equal to angle at water level):
$\tan(5^\circ) = \frac{127}{x}$
$x = \frac{127}{\tan(5^\circ)}$
$x \approx \frac{127}{0.0875} \approx 1451.43$ ft

Step3: Calculate distance to point B

Use tangent of $57^\circ$:
$\tan(57^\circ) = \frac{127}{y}$
$y = \frac{127}{\tan(57^\circ)}$
$y \approx \frac{127}{1.5399} \approx 82.47$ ft

Step4: Find distance A to B

Subtract the two horizontal distances:
$AB = x - y$
$AB \approx 1451.43 - 82.47$

Answer:

1368.9 feet