Question

a boat is heading towards a lighthouse, whose beacon-light is 129 feet above the water. from point a, the boats crew measures the angle of elevation to the beacon, $8^\circ$, before they draw closer. they measure the angle of elevation a second time from point b to be $13^\circ$. find the distance from point a to point b. round your answer to the nearest foot if necessary.
answer attempt 1 out of 2
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Explanation:

Step1: Define variables for distances

Let $AL$ = distance from A to lighthouse base, $BL$ = distance from B to lighthouse base, $AB = AL - BL$.

Step2: Calculate $AL$ using tangent

$\tan(8^\circ) = \frac{129}{AL}$
$AL = \frac{129}{\tan(8^\circ)}$
$AL \approx \frac{129}{0.1405} \approx 918.15$

Step3: Calculate $BL$ using tangent

$\tan(13^\circ) = \frac{129}{BL}$
$BL = \frac{129}{\tan(13^\circ)}$
$BL \approx \frac{129}{0.2309} \approx 558.70$

Step4: Compute distance $AB$

$AB = AL - BL$
$AB \approx 918.15 - 558.70 = 459.45 \approx 455$ (rounded to nearest foot)

Answer:

455 feet