Question

problem #3: a surveyor is trying to find the length of a lake. he finds a baseline distance ab of 125 m. angle a is measured to be 41.6° and angle b is measured at 124.3°. find the length of the lake d.

Explanation:

Step1: Calculate angle C

The sum of angles in a triangle is $180^\circ$.
$\angle C = 180^\circ - 41.6^\circ - 124.3^\circ = 14.1^\circ$

Step2: Apply Law of Sines

Relate sides and angles: $\frac{d}{\sin A} = \frac{AB}{\sin C}$

Step3: Solve for d

Rearrange to isolate d, substitute values.
$d = \frac{AB \cdot \sin A}{\sin C} = \frac{125 \cdot \sin(41.6^\circ)}{\sin(14.1^\circ)}$
Calculate sines: $\sin(41.6^\circ) \approx 0.664$, $\sin(14.1^\circ) \approx 0.244$
$d \approx \frac{125 \cdot 0.664}{0.244} \approx 338.52$

Answer:

The length of the lake $d$ is approximately 339 meters (rounded to the nearest whole number, or 338.5 meters for more precision).