Question

a boat is heading towards a lighthouse, whose beacon-light is 141 feet above the water. from point a, the boat’s crew measures the angle of elevation to the beacon, 7°, before they draw closer. they measure the angle of elevation a second time from point b at some later time to be 14°. find the distance from point a to point b. round your answer to the nearest foot if necessary.

Explanation:

Step1: Define variables for distances

Let $d_A$ = horizontal distance from A to lighthouse base, $d_B$ = horizontal distance from B to lighthouse base, height $h=141$ ft.

Step2: Calculate $d_A$ using tangent

Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$ for point A:
$\tan(7^\circ)=\frac{141}{d_A}$
$d_A=\frac{141}{\tan(7^\circ)}$
Calculate $\tan(7^\circ)\approx0.1228$, so $d_A\approx\frac{141}{0.1228}\approx1148.21$ ft

Step3: Calculate $d_B$ using tangent

Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$ for point B:
$\tan(14^\circ)=\frac{141}{d_B}$
$d_B=\frac{141}{\tan(14^\circ)}$
Calculate $\tan(14^\circ)\approx0.2493$, so $d_B\approx\frac{141}{0.2493}\approx565.58$ ft

Step4: Find distance $AB$

$AB = d_A - d_B$
$AB\approx1148.21 - 565.58$

Answer:

583 feet