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: Find angle C

In a triangle, the sum of interior angles is \(180^\circ\). So, \(\angle C=180^\circ - \angle A-\angle B\).
Substitute \(\angle A = 41.6^\circ\) and \(\angle B=124.3^\circ\) into the formula:
\(\angle C=180^\circ - 41.6^\circ- 124.3^\circ=14.1^\circ\)

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{d}{\sin B}=\frac{AB}{\sin C}\). We know \(AB = 125\space m\), \(\angle B=124.3^\circ\), \(\angle C = 14.1^\circ\). We need to solve for \(d\).
Rearranging the formula for \(d\), we get \(d=\frac{AB\times\sin B}{\sin C}\)

Step3: Calculate the values

First, calculate \(\sin(124.3^\circ)\) and \(\sin(14.1^\circ)\)
\(\sin(124.3^\circ)=\sin(180^\circ - 55.7^\circ)=\sin(55.7^\circ)\approx0.826\)
\(\sin(14.1^\circ)\approx0.244\)
Then substitute into the formula for \(d\):
\(d=\frac{125\times0.826}{0.244}\approx\frac{103.25}{0.244}\approx423.16\space m\)

Answer:

The length of the lake \(d\) is approximately \(423\space m\) (rounded to a reasonable decimal place)