Chapter 9.6, Problem 44PS

Solution

To find: The area of research triangle connecting three cities.

The area of research triangle is 62.8 sq. miles.

Given data:

The sides of the research triangle are distance from Raleigh to Durham 18 miles, from Raleigh to Chapel Hill 23 miles, from Chapel Hill to Durham 8 miles.

Method/Formula used:

  

The ‘Law of Cosines’ states for a triangle ABC , Fig. 1:

  $\cos \angle A=\frac{b^2+c^2-a^2}{2bc}$ … (1)

The area of triangle ABC is

  $\Delta =\frac{1}{2}bc\sin A$ … (2)

Calculations:

The research triangle ABC (say) with sides AB or c equals 23 miles, BC or $a$equals 8 miles and AC or b equals 18 miles, is shown in Fig.2.

  

Determine angle A , using result (1) as:

Substitute 8 for a , 18 for b and 23 for c in result (1),

  $\cos \angle A=\frac{b^2+c^2-a^2}{2bc}$

  $\begin{array}{c}\cos \angle A=\frac{{18}^2+{23}^2-8^2}{2\cdot \left(18\right)\cdot \left(23\right)}\\ \angle A={\cos }^{-1}\left(\frac{789}{828}\right)\\ =17.66°\end{array}$

Substitute $17.66°$ for $\angle A$ , 18 for b and 23 for c in result (2), area of research triangle ABC is

  $\begin{array}{c}\Delta ABC=\frac{1}{2}\left(18\right)\left(23\right)\sin \left(17.66°\right)\\ =62.8\end{array}$

Thus, the area of the Research triangle is 62.8 sq. miles.