Question

a surveyor measures the distance to the two landmarks, as shown in the diagram. what is the distance, d, between the landmarks?

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(d^{2}=a^{2}+b^{2}-2ab\cos C\). Here, \(a = 90\), \(b=130\), and \(C = 65^{\circ}\).

$$d^{2}=90^{2}+130^{2}-2\times90\times130\times\cos(65^{\circ})$$

Step2: Calculate each term

• \(90^{2}=8100\)
• \(130^{2}=16900\)
• \(\cos(65^{\circ})\approx0.4226\)
• \(2\times90\times130\times0.4226 = 2\times11700\times0.4226=23400\times0.4226 = 9988.84\)

$$d^{2}=8100 + 16900-9988.84$$

Step3: Simplify the right - hand side

\(8100+16900=25000\)
\(25000 - 9988.84=15011.16\)

$$d^{2}=15011.16$$

Step4: Solve for \(d\)

Take the square root of both sides: \(d=\sqrt{15011.16}\approx123\)

Answer:

\(123\ m\)