tow people begin walking from the same point at the same time. One walks due south at 3 mph and the other walks southeast at 4 mph. Approximately how far apart are the two after one hour walking?
use the law of cosines.
x^2 = 3^2 + 4^2 - 2(3)(4) cos 45°
To find the distance between the two people after one hour, we need to calculate the displacement of each person separately.
The person walking due south at 3 mph will have a displacement of 3 miles south after one hour. This displacement can be represented as a vector with a magnitude of 3 miles in the south direction.
The person walking southeast at 4 mph will have a displacement in both the south and east directions. To calculate the displacement in each direction, we can use the Pythagorean theorem.
The distance traveled in the south direction is given by speed × time = 4 mph × 1 hour = 4 miles.
The distance traveled in the east direction (since southeast is halfway between south and east) can be calculated using the same speed × time formula: 4 mph × 1 hour = 4 miles.
So, the person walking southeast has a displacement that can be represented as a vector with a magnitude of 4 miles in the east direction and 4 miles in the south direction.
Now, we can find the distance between the two points using the Pythagorean theorem:
distance^2 = (east displacement)^2 + (south displacement)^2
distance^2 = 4^2 + (3)^2
distance^2 = 16 + 9
distance^2 = 25
Taking the square root of both sides gives us:
distance = √25
distance = 5
Therefore, the two people will be approximately 5 miles apart after one hour of walking.
To calculate the distance between the two people after one hour, we need to find the positions of each person at that time. Let's break down the problem step-by-step:
1. Determine the distance the person walking due south would cover in one hour:
The person walks due south at a speed of 3 mph. Since speed is defined as distance divided by time, after one hour, the person would have covered a distance of 3 miles due south.
2. Determine the distance the person walking southeast would cover in one hour:
The person walks southeast at a speed of 4 mph. Since the direction is southeast, we can consider the person's motion as a combination of south (vertical) and east (horizontal) components. The ratio of the vertical (south) to horizontal (east) components is 1:1 because southeast is 45 degrees between south and east. Therefore, after one hour, the person would have covered 4 miles in the southeast direction.
3. Calculate the distance between the two people:
To find the distance between the two people, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the southward distance covered by the first person is one side of the triangle, and the southeast distance covered by the second person is another side. The distance between the two people, which is the hypotenuse of the right triangle formed by their paths, is the third side.
So, using the Pythagorean theorem, we can calculate the distance between the two people:
distance^2 = (3 miles)^2 + (4 miles)^2
distance^2 = 9 + 16
distance^2 = 25
distance = √25
distance = 5 miles
Therefore, after one hour of walking, the two people would be approximately 5 miles apart.