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°. at some later time, damian takes another measurement and finds the angle of depression to the boat (now at point b) to be 52°. 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

Explanation:

Step1: Find distance AL

Angle of depression = angle of elevation. Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$.
$\tan(21^\circ)=\frac{107}{AL} \implies AL=\frac{107}{\tan(21^\circ)}$
$AL\approx\frac{107}{0.3839}\approx278.7$ feet

Step2: Find distance BL

Use same tangent relationship for 52°.
$\tan(52^\circ)=\frac{107}{BL} \implies BL=\frac{107}{\tan(52^\circ)}$
$BL\approx\frac{107}{1.2799}\approx83.6$ feet

Step3: Calculate distance AB

Subtract BL from AL.
$AB=AL-BL\approx278.7-83.6=195.1$
Wait, correction: Recalculate with more precise tangent values:
$\tan(21^\circ)\approx0.383864035$, so $AL=\frac{107}{0.383864035}\approx278.74$
$\tan(52^\circ)\approx1.279941632$, so $BL=\frac{107}{1.279941632}\approx83.60$
$AB=278.74-83.60=195.14\approx195$ feet

Corrected final answer:

Step1: Calculate distance AL

Use tangent of 21°:
$AL=\frac{107}{\tan(21^\circ)}\approx\frac{107}{0.3839}\approx278.7$

Step2: Calculate distance BL

Use tangent of 52°:
$BL=\frac{107}{\tan(52^\circ)}\approx\frac{107}{1.280}\approx83.6$

Step3: Find AB by subtraction

$AB=278.7-83.6=195.1\approx195$

Answer:

223 feet