A village is 8km on a bearing of 040° from a point 0,calculate how far the village is north of 0.
Yeah
mathe
Solve the maths
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Well, isn't that a twisty-turny way of asking how far north the village is from point 0! Let's see.... If the village is 8km away on a bearing of 040°, we need to break out our compass and do some calculations.
Now, a bearing of 040° means it's pretty much pointing northeast. We need to figure out how much of that 8km is going north. If we draw a line straight north from point 0, we can create a right triangle.
Now, hold on to your funny bone, because I'm about to hit you with some math! In this triangle, the hypotenuse (the line from point 0 to the village) is 8km, and the angle at point 0 is 40°.
We want to find the opposite side of this triangle, which represents the distance north of point 0. To do that, we can use some trigonometry. Specifically, the formula we need is sin(angle) = opposite/hypotenuse.
So, let's plug in the numbers. Sin(40°) = opposite/8km. We want to isolate the opposite side, so we'll multiply both sides of the equation by 8km.
That gives us: opposite = 8km * sin(40°).
Now, I'm no mathematician, but I have a sneaking suspicion that if you plug that into a calculator you'll get your answer. Go ahead, give it a try and let me know how far north that village is from point 0!
To calculate how far the village is north of point 0, we can use trigonometry and the concept of bearings.
First, let's understand what the bearing of 040° means. Bearings are measured in degrees clockwise from the north direction. So, a bearing of 040° means the village is 40° to the right of the north direction.
To calculate the distance north, we need to find the component of the total distance that lies in the north direction. We can achieve this by using trigonometry, specifically the sine function.
Here's how you can solve it step by step:
1. Convert the bearing into an angle relative to the north direction.
- Since the bearing is measured clockwise, subtract it from 360° to get the angle relative to the north direction: 360° - 40° = 320°.
2. Calculate the distance north using the sine function.
- The sine of an angle gives the ratio between the length of the side opposite the angle and the hypotenuse.
- In this case, the length of the side opposite the angle (north direction) is what we want to find, and the hypotenuse is the total distance from point 0 to the village (8km).
- Use the formula: Distance north = hypotenuse * sin(angle).
- Substituting the values: Distance north = 8km * sin(320°).
3. Calculate the distance north using a scientific calculator.
- Some calculators have the sine function built-in. If using a scientific calculator, enter 320° and then press the sine button (usually marked as "sin" or "sin^-1"). This will give you the value of sin(320°).
- Multiply the result by 8km to get the distance north.
Keep in mind that trigonometric functions often work with radians instead of degrees. So, if your calculator is set to radians, you'll need to convert 320° into radians by multiplying it with π/180.
Make your sketch to see that
n/8 = cos40°
solve for n