Question

laura wants to know the length of a tunnel built through a mountain. to do so, she makes the measurements shown in the figure below. use these measurements to find the length of the tunnel.
carry your intermediate computations to at least four decimal places.
round your answer to the nearest tenth of a meter.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Here, \(a = 125\), \(b=270\), and \(C = 73^{\circ}\).
Substitute the values into the formula: \(c^{2}=125^{2}+270^{2}-2\times125\times270\times\cos(73^{\circ})\).
First, calculate \(125^{2}=15625\), \(270^{2}=72900\), and \(\cos(73^{\circ})\approx0.2924\).
Then \(2\times125\times270\times0.2924 = 625\times270\times0.2924= 625\times78.948 = 19737\).
\(125^{2}+270^{2}-2\times125\times270\times\cos(73^{\circ})=15625 + 72900-19737\).
\(15625+72900 = 88525\), and \(88525-19737=68788\).

Step2: Find the value of \(c\)

Take the square - root of \(68788\), \(c=\sqrt{68788}\approx262.3\).

Answer:

\(262.3\)