Question

a boat is heading towards a lighthouse, where damian is watching from a vertical distance of 107 feet above the water. damian measures an angle of depression to the boat at point $a$ to be $21^{\circ}$. at some later time, damian takes another measurement and finds the angle of depression to the boat (now at point $b$) to be $52^{\circ}$. find the distance from point $a$ to point $b$. round your answer to the nearest foot if necessary.
answer attempt 1 out of 2
feet
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Explanation:

Step1: Relate angles of depression

Angle of depression = angle of elevation from boat to Damian. So, $\angle BAL = 21^\circ$, $\angle BBL = 52^\circ$, height $h=107$ ft.

Step2: Calculate distance AL

Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. For point A:
$\tan(21^\circ)=\frac{107}{AL}$
$AL=\frac{107}{\tan(21^\circ)}$
$AL\approx\frac{107}{0.3839}\approx278.7$ ft

Step3: Calculate distance BL

For point B:
$\tan(52^\circ)=\frac{107}{BL}$
$BL=\frac{107}{\tan(52^\circ)}$
$BL\approx\frac{107}{1.2799}\approx83.6$ ft

Step4: Find distance AB

$AB=AL-BL$
$AB\approx278.7-83.6=195.1$
Wait, correction: Recalculated with more precise tan values:
$\tan(21^\circ)\approx0.383864$, so $AL=\frac{107}{0.383864}\approx278.74$
$\tan(52^\circ)\approx1.279942$, so $BL=\frac{107}{1.279942}\approx83.60$
$AB=278.74-83.60=195.14\approx195$ ft

Corrected final answer: 195 feet

(Revised):

Step1: Match angles of elevation

Angle of depression = angle of elevation. $\angle A=21^\circ$, $\angle B=52^\circ$, $h=107$ ft.

Step2: Compute distance from A to lighthouse

$\tan(21^\circ)=\frac{107}{AL}$
$AL=\frac{107}{\tan(21^\circ)}\approx278.7$ ft

Step3: Compute distance from B to lighthouse

$\tan(52^\circ)=\frac{107}{BL}$
$BL=\frac{107}{\tan(52^\circ)}\approx83.6$ ft

Step4: Calculate AB distance

$AB=AL-BL$
$AB\approx278.7-83.6=195$ ft

Answer:

219 feet