Question

question
a boat is heading towards a lighthouse, whose beacon-light is 115 feet above the water. from point a, the boats crew measures the angle of elevation to the beacon, 11°, before they draw closer. they measure the angle of elevation a second time from point b to be 24°. find the distance from point a to point b. round your answer to the nearest foot if necessary.
answer attempt 1 out of 2

Explanation:

Step1: Define variables for distances

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

Step2: Calculate $AL$ using tangent

Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$ for point $A$:
$\tan(11^\circ) = \frac{115}{AL}$
$AL = \frac{115}{\tan(11^\circ)}$
Calculate $\tan(11^\circ)\approx0.1944$, so $AL\approx\frac{115}{0.1944}\approx591.56$ feet.

Step3: Calculate $BL$ using tangent

Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$ for point $B$:
$\tan(24^\circ) = \frac{115}{BL}$
$BL = \frac{115}{\tan(24^\circ)}$
Calculate $\tan(24^\circ)\approx0.4452$, so $BL\approx\frac{115}{0.4452}\approx258.31$ feet.

Step4: Compute distance $AB$

Subtract $BL$ from $AL$:
$AB = AL - BL$
$AB\approx591.56 - 258.31 = 333.25$

Answer:

333 feet