Explanation
If the cable could went through the pond there would be only one straight section conecting the two points. If we draw this new section in the picture we'll form a triangle:
In order to find how much extra cable is needed because of the pond we must find the length of the imaginary cable that connects the points through the dot, for this length we are going to use x.
The cosine rule will help as find it. Let's assume that we have a triangle with an angle A which has an opposite side with a length a and the lengths of the other two sides are b and c. Then the cosine rule states the following:
[tex]a^2=b^2+c^2-2bc\cos A[/tex]
We can apply this to our triangle. The 145° angle that we know is A, its opposite side a is x and the remaining sides b and c are the two cable sections of 371 m and 440 m. Then we get:
[tex]\begin{gathered} x^2=371^2+440^2-2\cdot371\cdot440\cdot\cos145^{\circ} \\ x^2=598677.7594 \end{gathered}[/tex]
Then we apply a square root to both sides of this equation:
[tex]\begin{gathered} \sqrt{x^2}=\sqrt{598677.7594} \\ x=773.74 \end{gathered}[/tex]
So without the pond the length of the cable would have been of 773.74 m. In order to find the amount of extra cable needed we must take the total length of both sections and substract 773.74 m from it. Then we get:
[tex]371+440-773.74=37.26[/tex]Answer
Then the answer is 37.26m.
