The Christmas Day parade in Manhattan, New York started on 145th St. The people walked north on this road for 4 miles. Then they turned on 34th street and walked for three more miles. 34th street has a bearing of N46°E. How far is the parade from their starting point on 145th St. and what bearing should they take to get back?
To calculate the distance and bearing, we can use trigonometry and compass bearings. Let's break down the problem step by step.
Step 1: Calculate the east-west and north-south components of the displacement.
The parade walked 4 miles north from 145th St. Since Manhattan has a rectangular grid street system, moving north implies movement in the positive y-direction. Therefore, their north-south displacement is +4 miles.
Next, the parade turned on 34th street. The given bearing N46°E means the direction is 46 degrees east of north. To calculate the east-west displacement, we need to decompose the displacement vector into its components using trigonometry.
East-West displacement = 3 miles * cos(46°)
North-South displacement = 3 miles * sin(46°)
Step 2: Calculate the total displacement.
The total north-south displacement is +4 miles. Adding the north-south component from the east-west displacement, we get:
Total north-south displacement = 4 miles + 3 miles * sin(46°)
Step 3: Calculate the total distance from the starting point.
Using the Pythagorean theorem, the total distance from the starting point can be calculated as follows:
Total distance = √(Total north-south displacement)^2 + (East-West displacement)^2
Step 4: Calculate the bearing to get back.
To calculate the bearing to get back to the starting point, we need to determine the angle between the east direction (positive x-axis) and the line connecting the starting point and the ending point. This can be done using inverse tangent (arctan).
Bearing = arctan(East-West displacement / Total north-south displacement)
Now, let's calculate the results:
East-West displacement = 3 miles * cos(46°) ≈ 2.018 miles
North-South displacement = 3 miles * sin(46°) ≈ 2.221 miles
Total north-south displacement = 4 miles + 2.221 miles ≈ 6.221 miles
Total distance = √(6.221 miles)^2 + (2.018 miles)^2 ≈ 6.628 miles
Bearing = arctan(2.018 miles / 6.221 miles) ≈ 17.949°
Thus, the parade is approximately 6.628 miles away from their starting point on 145th St., and they should take a bearing of approximately 17.949° to get back.