AP Physics Problem - Highway between two cities

[zud3652]

Homework Statement

A highway is to be built between two towns, one of which lies 40.7 km south and 72.5 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?

Homework Equations

No equations are given. I used Pythagorean theorem.

The Attempt at a Solution

I got 83.14m for the first part. I got 29.31 degrees for the second part but I don't think this is right.

THANK YOU SO MUCH FOR YOUR HELP!

PICTURE: https://www.dropbox.com/s/052vjjh3by6kbnl/Untitled.png?dl=0

 

[Matterwave]

Are you allowed to assume the Earth is flat? I'm guessing you are, otherwise this become a spherical geometry problem.

How did you arrive at your answers? Did you draw a picture?

 

[zud3652]

Yes we can assume the Earth is flat. The first part is simple, just Pythagorean theorem, really my question is about part B. I am confused when it says "with respect to due west". I added a picture to the post to show you what I have.

 

[Matterwave]

Ah, reading this question again, that is a very ambiguous way to phrase the question b, since it does not tell you which city is to serve as the origin.

There are two ways to interpret this question. Either it's asking for angle ABC or or angle 180°-ABC.

I don't think I can decipher which the question is asking for. Could you perhaps ask an instructor?

 

[HalfLostAndSearching]

I would like to commend you on successfully using the Pythagorean theorem to solve this problem. Your answer of 83.14 km for the shortest length of highway is correct. However, your answer of 29.31 degrees for the angle is not correct.

To find the angle, we can use trigonometric ratios. Since we know the lengths of the sides of the triangle (40.7 km and 72.5 km), we can use the tangent function to find the angle. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is 40.7 km and the adjacent side is 72.5 km.

So, we can write the equation: tan(theta) = 40.7/72.5

Using a calculator, we can find the inverse tangent (or arctangent) of 40.7/72.5 to be approximately 28.97 degrees. However, this angle is measured from the positive x-axis, not the positive y-axis. To find the angle with respect to due west, we need to subtract this angle from 90 degrees (since due west is perpendicular to due north). So, the final answer for the angle would be 90 - 28.97 = 61.03 degrees.

Therefore, the shortest length of highway that can be built between the two towns is 83.14 km, and the angle of the highway with respect to due west is 61.03 degrees.

I hope this helps. Keep up the good work!