Question

a boat is heading towards a lighthouse, whose beacon-light is 148 feet above the water. from point a, the boats crew measures the angle of elevation to the beacon, $6^\circ$, before they draw closer. they measure the angle of elevation a second time from point b at some later time to be $23^\circ$.
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: Define variables for distances

Let $x$ = horizontal distance from point $A$ to base of lighthouse, $y$ = horizontal distance from point $B$ to base of lighthouse. Height of lighthouse $h=148$ ft.

Step2: Calculate distance from A to base

Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$ for point A:
$\tan(6^\circ)=\frac{148}{x}$
$x=\frac{148}{\tan(6^\circ)}$
Calculate $\tan(6^\circ)\approx0.1051$, so $x\approx\frac{148}{0.1051}\approx1408.18$ ft

Step3: Calculate distance from B to base

Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$ for point B:
$\tan(23^\circ)=\frac{148}{y}$
$y=\frac{148}{\tan(23^\circ)}$
Calculate $\tan(23^\circ)\approx0.4245$, so $y\approx\frac{148}{0.4245}\approx348.64$ ft

Step4: Find distance A to B

Subtract the two distances: $AB = x - y$
$AB\approx1408.18 - 348.64$

Answer:

1059 feet