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

Answer 1

In general, the law of cosines states that

`\begin{gathered} c^2=a^2+b^2-2ab\cos C \\ a,b,c\rightarrow\text{ sides of a triangle} \\ C\rightarrow\text{ opposite angle to side c} \end{gathered}`

Thus, in our case,

`\begin{gathered} AB^2=BC^2+CA^2-2*BC*CA\cos(37.4\degree) \\ \Rightarrow AB^2=2.45^2+3.16^2-2*2.45*3.16\cos(37.4\degree) \\ \Rightarrow AB^2=3.68738... \\ \Rightarrow AB\approx1.92 \end{gathered}`

Therefore, the answer is 1.92mi