A mountain biker rides a given distance due north, then drives twice that far to the west. At the end of the trip the direct distance from the starting point is 112meters. What is the length of each leg of the trip and in which direction is the end point?
d^2 = (2n)^2 + n^2 = 112^2
4n^2 + n^2 = 12,544
5n^2 = 12544
n^2 = 2508.8
n = 50 m. = North leg.
2n = 100 m. = West leg.
tan A = 2n/n = 2.0
A = 63.43o West of North = Direction.
To solve this problem, we can use the Pythagorean theorem and basic trigonometry. Let's break down the problem step by step:
1. Let's assume that the distance the mountain biker rides due north is 'x' meters.
2. The distance the mountain biker drives twice that far to the west would be '2x' meters.
3. Now, let's apply the Pythagorean theorem. According to the theorem, the square of the hypotenuse (the direct distance from the starting point) is equal to the sum of the squares of the other two sides.
So, we have:
Hypotenuse^2 = (x^2 + (2x)^2)
4. Simplifying the equation:
Hypotenuse^2 = x^2 + 4x^2
Hypotenuse^2 = 5x^2
5. Taking the square root of both sides to solve for Hypotenuse:
Hypotenuse = √(5x^2)
Hypotenuse = x√5
6. We are given that the Hypotenuse is equal to 112 meters. Plugging in this value:
112 = x√5
7. Solving for 'x':
x = 112 / √5
8. Simplifying this expression:
x ≈ 49.98 meters
9. Now that we know the value of 'x', we can find the length of each leg of the trip. The distance the mountain biker rides due north is approximately 49.98 meters, and the distance driven to the west is twice that, which is approximately 99.96 meters.
10. To determine the direction of the endpoint, we use basic trigonometry. The angle formed between the direct distance from the starting point and the distance ridden due north is given by:
Angle = arctan(49.98 / 99.96)
11. Using a calculator:
Angle ≈ 26.6 degrees
12. Therefore, the endpoint is located approximately 112 meters away from the starting point, at an angle of approximately 26.6 degrees west of due north.
So, the length of each leg of the trip is approximately 49.98 meters, and the endpoint is in the northwest direction (26.6 degrees west of due north).