A small map shows New York to be 486 miles in a direction 8° north of east from Columbus. The same map shows that Buffalo is 302 miles in a direction 56° west of north from New York. Assume a flat Earth, and use the given information to find the displacement from Columbus to Buffalo.
magnitude___miles
direction___degrees, north of east of columbus
I don't know what you have been taught. If you make a sketch, put the vectors head to tail, you have a triangle with two known legs, with the included angle (64deg) known. You can find the other side with the law of cosines.
Then, with the law of sines, find the angle of it (with very little geometry).
Or, you can do it graphically.
Or, you can break up each vector into N and E components, then add both vectors, and then put them back together.
I would use the law of cosines, sines.
To find the displacement from Columbus to Buffalo, we need to calculate the magnitude and direction.
Let's start by breaking down the information given step-by-step:
Step 1: From Columbus to New York
- The map shows a distance of 486 miles.
- The direction is 8° north of east.
Step 2: From New York to Buffalo
- The map shows a distance of 302 miles.
- The direction is 56° west of north.
Now, let's proceed with the calculation:
Step 1: From Columbus to New York
Since the direction is given as north of east, we can treat it as a positive angle measured clockwise from the east direction.
To find the east component, we use the formula:
East Component = Magnitude * cos(Direction)
East Component = 486 miles * cos(8°)
East Component = 486 miles * 0.9903
East Component ≈ 481.03 miles (rounded to two decimal places)
To find the north component, we use the formula:
North Component = Magnitude * sin(Direction)
North Component = 486 miles * sin(8°)
North Component = 486 miles * 0.1389
North Component ≈ 67.42 miles (rounded to two decimal places)
Step 2: From New York to Buffalo
Since the direction is given as west of north, we can treat it as a positive angle measured clockwise from the north direction.
To find the north component, we use the formula:
North Component = Magnitude * cos(Direction)
North Component = 302 miles * cos(56°)
North Component = 302 miles * 0.5592
North Component ≈ 168.97 miles (rounded to two decimal places)
To find the west component, we use the formula:
West Component = Magnitude * sin(Direction)
West Component = 302 miles * sin(56°)
West Component = 302 miles * 0.8290
West Component ≈ 250.76 miles (rounded to two decimal places)
Finally, let's find the displacement from Columbus to Buffalo by adding the respective components:
East Component = 481.03 miles
North Component = 67.42 miles
Add East and West Components:
481.03 miles + (-250.76 miles) = 230.27 miles (rounded to two decimal places)
Add North Components:
67.42 miles + 168.97 miles = 236.39 miles (rounded to two decimal places)
The magnitude of the displacement from Columbus to Buffalo is approximately 236.39 miles.
The direction of the displacement from Columbus to Buffalo is north of east of Columbus by approximately 230.27 degrees.
To find the displacement from Columbus to Buffalo, we can break down the problem into two components: the north-south component and the east-west component.
First, let's find the north-south component. Given that Buffalo is 302 miles in a direction 56° west of north from New York, we can apply simple trigonometry:
North-South Component = 302 * sin(56°)
Next, let's find the east-west component. Given that New York is 486 miles in a direction 8° north of east from Columbus, we can again apply trigonometry:
East-West Component = 486 * cos(8°)
Now, we can find the magnitude of the displacement using the Pythagorean theorem:
Magnitude of Displacement = √(North-South Component² + East-West Component²)
Finally, the direction of the displacement can be found using the inverse tangent (arctan) function:
Direction = arctan(North-South Component / East-West Component)
Plugging in the values and solving the equations will give us the answer:
North-South Component = 302 * sin(56°) = 244.66 miles
East-West Component = 486 * cos(8°) = 481.25 miles
Magnitude of Displacement = √(244.66² + 481.25²) = 537.89 miles (approximately)
Direction = arctan(244.66 / 481.25) = 27.26° (approximately)
Therefore, the displacement from Columbus to Buffalo is approximately 537.89 miles, in a direction 27.26° north of east from Columbus.