Question

two checkpoints, a and b, in an orienteering course are 5 km apart. another checkpoint, c, is 3 km from checkpoint a. the angle between ab and ac is 25°. determine the distance between checkpoints b and c.

Explanation:

Step1: Identify the formula

We use the Law of Cosines. For a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite to side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). Let \(AB = c = 5\), \(AC = b=3\), \(\angle BAC=\theta = 25^{\circ}\), and we want to find \(BC=a\).

Step2: Substitute values into the formula

\[a^{2}=3^{2}+5^{2}-2\times3\times5\times\cos(25^{\circ})\]
\[a^{2}=9 + 25-30\times\cos(25^{\circ})\]
We know that \(\cos(25^{\circ})\approx0.9063\).
\[a^{2}=9 + 25-30\times0.9063\]
\[a^{2}=34 - 27.189\]
\[a^{2}=6.811\]

Step3: Solve for \(a\)

\[a=\sqrt{6.811}\approx 2.61\] (rounded to two - decimal places)

Answer:

\(2.61\) km