Question

a boat is heading towards a lighthouse, whose beacon-light is 107 feet above the water. from point a, the boats crew measures the angle of elevation to the beacon, $5^\circ$, before they draw closer. they measure the angle of elevation a second time from point b to be $11^\circ$. find the distance from point a to point b. round your answer to the nearest foot if necessary.
answer attempt 2 out of 2
feet

Explanation:

Step1: Find distance AL

Use tangent for angle at A:
$\tan(5^\circ) = \frac{107}{AL}$
$AL = \frac{107}{\tan(5^\circ)}$
$AL \approx \frac{107}{0.0875} \approx 1222.86$ feet

Step2: Find distance BL

Use tangent for angle at B:
$\tan(11^\circ) = \frac{107}{BL}$
$BL = \frac{107}{\tan(11^\circ)}$
$BL \approx \frac{107}{0.1944} \approx 550.41$ feet

Step3: Calculate AB distance

Subtract BL from AL:
$AB = AL - BL$
$AB \approx 1222.86 - 550.41 = 572.45$
Rounded to nearest foot: $AB \approx 572$ feet

*Note: Using more precise tangent values: $\tan(5^\circ)\approx0.0874887$, $\tan(11^\circ)\approx0.1943803$
$AL=\frac{107}{0.0874887}\approx1222.90$, $BL=\frac{107}{0.1943803}\approx550.46$
$AB=1222.90-550.46=672.44\approx672$ feet. The most accurate rounded value is 672 feet.*

Corrected precise calculation:

Step1: Find AL with precise tan

$AL = \frac{107}{\tan(5^\circ)} \approx \frac{107}{0.0874886635} \approx 1222.90$

Step2: Find BL with precise tan

$BL = \frac{107}{\tan(11^\circ)} \approx \frac{107}{0.1943803091} \approx 550.46$

Step3: Compute AB

$AB = 1222.90 - 550.46 = 672.44 \approx 672$

Final corrected answer: 672 feet

Answer:

596 feet