A person desires to reach a point that is 2.87 km from her present location and in a direction that is 47.5° north of east. However, she must travel along streets that are oriented either north-south or east-west. What is the minimum distance she could travel to reach her destination?
N=2.87Sin47.5
E=2.87 cos47.5
add the distances.
To find the minimum distance the person could travel to reach their destination, we can break down their movement into north-south and east-west components.
Let's start by drawing a diagram to visualize the situation. Assume the person's starting point is at the origin (0,0) on a coordinate plane.
The distance traveling north-south can be calculated using the sine function, while the distance traveling east-west can be calculated using the cosine function.
1. First, let's find the north-south distance:
- The angle is north of east, which means it is measured clockwise from the positive x-axis.
- To find the north-south component, we need to find the length of the side opposite the angle.
- Using the formula: north-south distance = (distance to destination) * sin(angle)
- Substituting the given values: north-south distance = 2.87 km * sin(47.5°)
2. Next, we'll find the east-west distance:
- Again, using the angle north of east, we need to find the length of the side adjacent to the angle.
- Using the formula: east-west distance = (distance to destination) * cos(angle)
- Substituting the given values: east-west distance = 2.87 km * cos(47.5°)
3. Finally, we can find the minimum distance by calculating the direct distance between the starting point and the destination.
- Using the Pythagorean theorem: minimum distance = sqrt((north-south distance)^2 + (east-west distance)^2)
Now, let's do the calculations:
north-south distance = 2.87 km * sin(47.5°) ≈ 2.87 km * 0.714 ≈ 2.05 km
east-west distance = 2.87 km * cos(47.5°) ≈ 2.87 km * 0.700 ≈ 2.01 km
minimum distance = sqrt((2.05 km)^2 + (2.01 km)^2) ≈ sqrt(4.20 km^2) ≈ 2.05 km
Therefore, the minimum distance the person could travel to reach their destination is approximately 2.05 km.