Chapter 12.5, Problem 27PPS

To determine

To find: Area of the land to the nearest square yard.

Answer to Problem 27PPS

Area of the land is $13098y{d}^2$

Explanation of Solution

Given information:

Let

  $\begin{array}{l}a=300\\ b=210\\ c=140\end{array}$

Using law of Cosines we get

  $a^2=b^2+c^2-2bc\cos A$

On substituting values

We get

  $\begin{array}{l}{a}^2=b^2+c^2-2bc\cos A\\ \Rightarrow {\left(300\right)}^2={\left(210\right)}^2+{\left(140\right)}^2-2\left(210\right)\left(140\right)\cos A\\ \Rightarrow 90000=44100+19600-58800\cos A\\ \Rightarrow 58800\cos A=63700-90000\\ \Rightarrow 58800\cos A=-26300\\ \Rightarrow \cos A=\frac{-26300}{58800}\\ \Rightarrow \cos A=-0.447\\ \Rightarrow A={\cos }^{-1}\left(-0.447\right)\\ \Rightarrow A={117}^0\end{array}$

We know

Area $=\frac{1}{2}bc\sin A$

On substituting values

We get

  $\Rightarrow$ Area $=\frac{1}{2}\left(210\right)\left(140\right)\sin {117}^0$

  $\Rightarrow$ Area $=\frac{1}{2}\left(210\right)\left(140\right)\left(0.891\right)$

  $\Rightarrow$ Area $=\frac{26195.6}{2}$

  $\Rightarrow$ Area $=13098$

Therefore,

Area of the land is $13098y{d}^2$