Chapter 6, Problem 40CR

To determine

To calculate:

Distance between $A$ and $C$ .

Answer to Problem 40CR

Distance between $A$ and $C$ is $615.1m$ .

Explanation of Solution

Given information:

  

Calculation:

Law of sines formula states,

  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

Law of cosine formula says hat in case $ABC$ is an Oblique triangle with sides $a,b,c$ then,

  $\begin{array}{l}{a}^2=b^2+c^2-2bc\cos A\\ b^2=a^2+c^2-2ac\cos B\\ c^2=a^2+b^2-2ab\cos C\end{array}$

Consider,

  $AB=425m,BC=300m$ and their included angle $\angle ABC=180°-65°=115°$

To find distance $AC$ ,

In $\Delta COD$ using cosine formula,

  $\begin{array}{l}A{C}^2=A{B}^2+B{C}^2-2\left(AB\right)\left(BC\right)\cos 115°\\ ={\left(425\right)}^2+{\left(300\right)}^2-2\left(425\right)\left(300\right)\left(-0.4226\right)\\ =180625+9000+107.763\\ =378388\\ AC=\sqrt{378388}\\ =615.132\\ =615.1\end{array}$

Therefore, Distance between $A$ and $C$ is $615.1m$ .