A man walks one mile east, then he walks one mile northeast and then he walks one mile east. How far is he from his initial position?
Consider east = +x, North = +y
(1,0) + (.707,.701) + (1,0)
=(2.707,0.707)
distance = 7.828
To find out how far the man is from his initial position after walking one mile east, one mile northeast, and one mile east, we can break it down step-by-step:
Step 1: Walking one mile east moves him one mile away from his initial position.
Step 2: Walking one mile northeast forms a right-angled triangle. In a right-angled triangle, if two sides are equal, the opposite angle is 45 degrees. So, in this case, the angle formed by walking one mile northeast is 45 degrees.
Step 3: Now, we can apply trigonometry to find out how far he is from his initial position after the second step. Using the cosine function, we can calculate the horizontal component of the triangle, which is adjacent to the 45-degree angle.
Adjacent side length = hypotenuse length * cosine(angle)
Adjacent side length = 1 mile * cosine(45 degrees) = 0.707 miles (rounded to three decimal places)
Step 4: Finally, walking one mile east again moves him one mile away from his current position.
So, to find out how far he is from his initial position, we add up the distances covered in each step:
Distance from the initial position = 1 mile (first step) + 0.707 miles (second step) + 1 mile (third step)
Therefore, the man is approximately 2.707 miles away from his initial position (rounded to three decimal places).
To find out how far the man is from his initial position, we need to use some basic geometry and trigonometry. Let's break down the steps:
1. The man walks one mile east. This means that he moves directly to the right along the horizontal axis. Since he moves straight east, his position along the vertical axis remains the same. Therefore, his new position is one mile to the right (east) of his initial position.
2. The man walks one mile northeast. This means that he moves in a diagonal direction, somewhere between east and north. To calculate the actual distance he covers along the diagonal, we can use the Pythagorean theorem. Since he already moved one mile east, his eastward distance is one mile. Let's assume his northward distance is x miles. According to the Pythagorean theorem, the square of the hypotenuse (in this case, the distance traveled) is equal to the sum of the squares of the other two sides. Therefore, using the theorem, we have:
(1 mile)^2 + x^2 = (distance)^2
Simplifying this equation, we get:
1 + x^2 = (distance)^2
Since we know the man traveled one mile northeast, the distance is equal to 1 mile. So we have:
1 + x^2 = 1^2
1 + x^2 = 1
x^2 = 0
x = 0
This means that the man did not move northward at all. Therefore, his position after walking one mile northeast is one mile east from his previous position.
3. Finally, the man walks one mile east again. Since he already moved one mile east, his new position is one mile to the east of his previous position.
Putting it all together, the man is two miles east of his initial position.