A highway is to be built between two towns, one of which lies 33.5 km south...

A highway is to be built between two towns, one of which lies 33.5 km south and 70.8 km west of the other. (a) What is the shortest length of highway that can be built between the two towns, and (b) at what angle would this highway be directed, as a positive angle with respect to due west?

Expert Solution

Call the starting town Town A, and label it with a point called A. Now, draw a line straight, south of this point. The length of the line is 33.5. Draw a line from the end of the line you just drew. It should be going west, and its length is 70.8. The end of the line you just drew is the second town, Town B. What we have is a right triangle. The shortest distance is always the straight line distance. We have the other 2 sides of the right triangle. Using thePythagorean theorem we get:

a^2 + b^2 = c^2
33.5^2 + 70.8^2 = c^2
c = 78.3255km

78.3255km is shortest length highway between the towns.

What angle with respect to west. This means the angle between the line going west and the hypotenuse. I'll call that angle theta.

Therefore, tan theta = opposite/adjacent = 33.5/70.8
theta = arctan(33.5/70.8) Note: arctan means inverse tan (tan^-1 on your calculator)
theta = 25.32degrees south of west

genius_generous answered 2 years ago

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