Let's solve the problem step-by-step:
1. Understand the Problem:
Bailey first rides 2 miles east and then 4 miles south. We need to determine the direct path to Marco's house, both the distance and the direction in compass terms.
2. Visualize the Scenario:
- Start at the origin point (0,0).
- Move 2 miles east, reaching the point (2, 0).
- Move 4 miles south, reaching the point (2, -4).
3. Determine the Direct Distance:
We can model this situation using a right triangle:
- One leg of the triangle represents the eastward travel distance (2 miles).
- The other leg of the triangle represents the southward travel distance (4 miles).
- The hypotenuse of the triangle represents the direct distance from Bailey's house to Marco's house.
Using the Pythagorean theorem:
[tex]\[
\text{Distance} = \sqrt{(2^2) + (4^2)} = \sqrt{4 + 16} = \sqrt{20} \approx 4.472 \text{ miles}
\][/tex]
4. Determine the Direction:
To find the direction, we use the angle formed by the hypotenuse with the eastward axis:
- The angle θ can be found using the arctangent function:
[tex]\[
\theta = \arctan\left(\frac{\text{southward distance}}{\text{eastward distance}}\right) = \arctan\left(\frac{4}{2}\right)
\][/tex]
[tex]\[
\theta \approx 63.435^\circ \text{ (since we have southward 4 miles and eastward 2 miles)}
\][/tex]
5. Convert the Angle into Compass Direction:
Since the angle is measured clockwise from east and the travel direction is south of east, the resulting compass direction can be described as:
- "63.435° south of east"
6. Summarize the Result:
- The direct distance from Bailey's house to Marco's house is approximately 4.472 miles.
- The direction Bailey would have travelled directly is approximately 63.435° south of east.
Thus, if Bailey could have directly travelled from his house to Marco's, he would have travelled approximately 4.472 miles in a direction of approximately 63.435° south of east.
