Chapter MPS, Problem 13.7MPS

To determine

Approximate the length of the lake.

Answer to Problem 13.7MPS

  $422.104yards.$

Explanation of Solution

Given information:

A surveyor starts at one end of the lake and walks $245$ yards to approximate the length of a quarry. He then turn $110°$ and walks $270$ yards until he arrives at the other end of the lake.

Calculation:

Consider the figure:

  

Suppose that the man starts from point $B$ and after moving $245$ yards towards $A$ he turns $110°$ and walks $270$ yards to reach the other end of the lake.

The law of Cosines is as follows:

  $a^2=b^2+c^2-2bc\cos A.$

Use law of Cosines and substitute $b=270,c=245,andA=110°$ .

  $\begin{array}{l}{a}^2=b^2+c^2-2bc\cos A\\ a^2={\left(270\right)}^2+{\left(245\right)}^2-2\times 270\times 245\cos 110°\\ =72900+60025-132300\times \left(-0.342\right)\\ =132925+45246.6\\ a^2=178171.6\\ a=\pm \sqrt{178171.6}\\ a=\pm 422.104\end{array}$

So the length cannot be negative quantity, therefore $a\approx \mathrm{422.104.}$

Hence, the approximated length of the lake is $422.104yards.$