Chapter 6.1, Problem 64E

To determine

To find: the length of bridge if it is built through point C mentioned in the figure.

Answer to Problem 64E

  $172.7\text{ yards}$

Explanation of Solution

Given information:

The figure given is as follows:

  

  $\begin{array}{l}AB=560\text{ yards}\\ \angle A=27°\\ \angle B=38°\end{array}$

Calculation:

Evaluate the value of $\angle C$ by using angle sum property of triangles as shown below.

  $\begin{array}{c}\angle A+\angle B+\angle C=180°\\ 27°+38°+\angle C=180°\left[\text{Put the values of}\angle A\text{ and}\angle B\right]\\ \angle C=180°-27°-38°\\ \angle C=115°\end{array}$

Use law of sine to evaluate BC as follows:

  $\begin{array}{c}\frac{BC}{\sin A}=\frac{AB}{\sin C}\\ \frac{BC}{\sin 27°}=\frac{560}{\sin 115°}\left[\text{Put }AB\text{, }A\text{ and }C\right]\\ BC=\left(\sin 27°\right)\left(\frac{560}{\sin 115°}\right)\\ =\left(0.4539\right)\left(\frac{560}{0.90630}\right)\\ =280.5\text{ yard}\end{array}$

Let $x$ be the bridge length, which can be evaluated as follows:

  $\begin{array}{c}\sin B=\frac{x}{BC}\\ \sin 38°=\frac{x}{280.52}\left[\text{Put }\angle \text{B and }BC\right]\\ x=\sin 38°\left(280.52\right)\\ =\left(0.61566\right)\left(280.52\right)\\ =172.7\text{ yards}\end{array}$

Thus the area of triangle is $172.7\text{ yards}$ .