Question

to find the distance across a small lake, a surveyor has taken the measurements shown. find the distance across the lake using this information.

note: the triangle is not drawn to scale.

distance = mi

enter your answer as a number; your answer should be accurate to 2 decimal places.

question help: worked example 1

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and angle \(C\) opposite side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos C\). In \(\triangle ABC\), let \(a = 1.17\), \(b = 2.81\), and \(C=42.3^{\circ}\). We want to find the length of side \(AB\) (let's call it \(x\)). Then \(x^{2}=1.17^{2}+2.81^{2}-2\times1.17\times2.81\times\cos(42.3^{\circ})\).
First, calculate \(1.17^{2}=1.3689\), \(2.81^{2}=7.8961\).
Next, calculate \(2\times1.17\times2.81 = 6.5514\), and \(\cos(42.3^{\circ})\approx0.74\).
Then \(6.5514\times0.74 = 4.848036\).

Step2: Compute \(x^{2}\)

\(x^{2}=1.3689 + 7.8961-4.848036\).
\(x^{2}=9.265 - 4.848036=4.416964\).

Step3: Find \(x\)

Take the square - root of \(x^{2}\). \(x=\sqrt{4.416964}\approx2.10\).

Answer:

\(2.10\)