Chapter 5.FOM, Problem 1P

Completing the Map Find the distance between the church and the City Hall.

To determine

To find:

The distance between the church and the City Hall.

Answer to Problem 1P

Solution:

The distance between the church and the City Hall is 1.41 mi.

Explanation of Solution

Surveying is a method of land measurement used for mapmaking.

Surveyors use a process called Triangulation in which a network of thousands of interlocking triangles is created on the area to be mapped.

The process is started by measuring the lengths of a baseline between two stations and the angle between these two stations and a third station are measured.

The Law of Sines is then used to calculate the other side of the triangles.

Law of sine:

The Law of Sines says that in any triangle the lengths of the sides are proportional to the sines of the corresponding opposite angles.

Let consider a triangle as A, B and C and the lengths of the corresponding opposite sides as a, b and c respectively.

The Law of Sines for the triangle ABC is, $\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

Calculation:

The Map of the Hometown is,

Let the distance between the church and the city hall be x.

The distance between the landmarks can be found by using the Law of Sines.

To find x, apply the Law of Sines to the triangle with vertices at City Hall, first bridge and the church.

$\begin{array}{rll}\frac{0.86}{\sin 30°}& =& \frac{x}{\sin 125°}\\ x& =& \frac{0.86\times \sin 125°}{\sin 30°}\\ x& =& \frac{0.86\times 0.8192}{0.5}\\ x& \approx & 1.4090\approx 1.41\text{ mi}\end{array}$

The distance between the church and the City Hall is 1.41 mi.