Chapter 12, Problem 39SGR

To determine

To find: How much fencing will the farmed need.

Answer to Problem 39SGR

Total fence farmer needs $=750.5$ feet.

Explanation of Solution

Given information:

Two sides of the triangular field have lengths of $120$ feet and $325$ feet

Let those two sides be $a$ and $b$

Measure of angle between those two sides is ${70}^0$

Let third side be $C$

Then

  $\begin{array}{l}C={70}^0\\ a=120\\ b=325\end{array}$

Given

  $\begin{array}{l}C={70}^0\\ a=120\\ b=325\end{array}$

The triangle should be solved with the Law of cosines

We know

  $\cos C=\frac{a^2+b^2-c^2}{2ab}$

On substituting values

We get

  $\begin{array}{l}\cos C=\frac{a^2+b^2-c^2}{2ab}\\ \Rightarrow \cos {70}^0=\frac{{120}^2+{325}^2-c^2}{2\left(120\right)\left(325\right)}\\ \Rightarrow 0.342=\frac{14400+105625-c^2}{78000}\\ \Rightarrow 14400+105625-c^2=78000\left(0.342\right)\\ \Rightarrow 120025-c^2=26676\\ \Rightarrow c^2=120025-26676\\ \Rightarrow c^2=93349\\ \Rightarrow c=\sqrt{93349}\\ \Rightarrow c=305.5\end{array}$

Therefore,

Third side of the field is $305.5$ feet

Then total length of the fence is

  $\begin{array}{l}Total=120+325+305.5\\ \Rightarrow Total=750.5\end{array}$

Therefore,

Total fence farmer needs $=750.5$ feet.