In an orienteering race, you walk 100 m due east and then walk N70¡ÆE for 60 m. How far are you from your
starting position, and at what bearing?
follow the method I showed you in the two previous posts.
you could also make a sketch and solve it using basic trig
I see a triangle with sides 100 and 60 and the contained angle of 160°
by the cosine law
x^2 = 100^2+60^2-2(100)(60)cos120
....
To find the distance and bearing from the starting position, we can use trigonometry.
First, let's break down the walks into their components.
The first walk due east is 100 m in that direction, which we can denote as a vector (100, 0).
The second walk is at an angle of N70°E, which means we need to split it into its north and east components.
To find the east component, we use the formula:
east component = distance * cos(angle)
Substituting the values, we have:
east component = 60 * cos(70°)
Calculating this, we find:
east component ≈ 60 * 0.3420 ≈ 20.52 m
To find the north component, we use the formula:
north component = distance * sin(angle)
Substituting the values, we have:
north component = 60 * sin(70°)
Calculating this, we find:
north component ≈ 60 * 0.9397 ≈ 56.38 m
Now, we can add up the east and north components to find the total displacement from the starting position:
total east displacement = 100 + 20.52 = 120.52 m
total north displacement = 0 + 56.38 = 56.38 m
Using the Pythagorean theorem, we can calculate the distance from the starting position:
distance = sqrt((total east displacement)^2 + (total north displacement)^2)
distance = sqrt((120.52)^2 + (56.38)^2)
distance ≈ sqrt(14551.3504 + 3179.2644)
distance ≈ sqrt(17730.6148)
distance ≈ 133.16 m (rounded to two decimal places)
To find the bearing, we use the formula:
bearing = atan(north component / east component)
Substituting the values, we have:
bearing ≈ atan(56.38 / 120.52)
Calculating this, we find:
bearing ≈ atan(0.4674)
bearing ≈ 25.66°
Therefore, the distance from the starting position is approximately 133.16 m, and the bearing is approximately 25.66°.
To find the distance from the starting position, you can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Given that you walked 100 m due east, and then N70¡ÆE (northeast), we can break down the distance traveled into east and north components.
The east component is 100 m, and the north component can be found by using the sine function. The sine of 70 degrees can be calculated by using a calculator or a trigonometric table. Let's assume it is 0.9397.
The north component is then calculated by multiplying the sine value by the distance traveled, which gives us 60 * 0.9397 = 56.38 m.
Now we have the east and north components of your movement. To find the distance from the starting position, we can use the Pythagorean theorem:
Distance^2 = East^2 + North^2
Distance^2 = 100^2 + 56.38^2
Distance^2 = 10000 + 3176.3844
Distance^2 = 13176.3844
Distance ≈ √13176.3844
Distance ≈ 114.81 m (rounded to two decimal places)
To find the bearing, we can use the tangent function. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the tangent of the bearing angle can be calculated using the north and east components:
Tangent(bearing) = North / East
Tangent(bearing) = 56.38 / 100
Tangent(bearing) = 0.5638
To find the bearing, we need to take the arctangent of this value (using a calculator or trigonometric table):
Bearing = Arctan(0.5638)
Bearing ≈ 29.17° (rounded to two decimal places)
Therefore, you are approximately 114.81 m away from your starting position at a bearing of approximately 29.17°.