A boy walks 60m on a bearing of 035 degree and then 300m due east. How far is he from his starting point?
Well, if the boy keeps walking in that direction, he might just end up discovering a new hidden treasure! But to answer your question, we can solve this using the magical powers of mathematics.
First, let's imagine the boy's starting point as the center of a circle. The 60m he walks on a bearing of 035 degrees can be visualized as an arc on that circle.
Now, the 300m due east is a line segment that starts at the end of that arc and keeps going straight. So, what we have is a beautiful isosceles triangle with two radii of the circle and the line segment between them.
To find how far the boy is from his starting point, we can use a bit of trigonometry magic. With the given bearing and distance, we can determine the angles of the triangle. From there, we can use the cosine rule to find the missing side length, which is the distance we're after.
But let me take a break from all this math talk! In the meantime, you can imagine the boy's adventure as a thrilling quest to find the lost treasure of Pythagoras!
To find the distance from the starting point, we can treat the boy's movement as a right-angled triangle.
Let's break down the steps:
1. The boy walks 60m on a bearing of 035 degrees. This creates a right-angled triangle, with the 035 degrees angle being the direction of his walk.
2. The adjacent side of the triangle represents the distance the boy walks towards the east, which is 300m.
3. The opposite side represents the distance the boy walks towards the north. To find this distance, we can use trigonometry. We can use the fact that tangent(θ) = opposite / adjacent.
Since the angle is 035 degrees and we know the adjacent side is 300m, we can rearrange this equation to solve for the opposite side:
opposite = tangent(035 degrees) * 300m
4. Calculate the value of tangent(035 degrees):
Using a calculator, we find that tangent(035 degrees) is approximately 0.7002.
5. Substitute the value of tangent(035 degrees) into the equation to find the opposite side:
opposite = 0.7002 * 300m ≈ 210.06m
6. Now, we can use the Pythagorean theorem to find the hypotenuse of the triangle, which represents the distance from the starting point. The Pythagorean theorem states that for a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
hypotenuse^2 = opposite^2 + adjacent^2
Substituting the values we found, we have:
hypotenuse^2 = 210.06^2 + 300^2
7. Solve for the hypotenuse:
hypotenuse^2 = 44,125.60 + 90,000
hypotenuse^2 = 134,125.60
Taking the square root of both sides:
hypotenuse ≈ √(134,125.60)
hypotenuse ≈ 366.22m
Therefore, the boy is approximately 366.22m away from his starting point.
To find the distance of the boy from his starting point, we can use the concept of vectors.
We can break down the boy's movement into two components: one in the north direction (bearing of 035 degrees) and one in the east direction (300m due east).
Using trigonometry, we can calculate the respective distances in each direction.
The north component can be calculated by taking the product of sin(θ) and the total distance covered:
North Component = 60m * sin(35°)
Next, we can calculate the east component by taking the product of cos(θ) and the total distance covered:
East Component = 60m * cos(35°)
By summing up the north and east components, we can determine the total displacement from the starting point:
Total Displacement = sqrt(North Component^2 + East Component^2)
Finally, we can substitute the values into the equations and solve for the total displacement:
North Component = 60m * sin(35°)
= 60m * 0.5736
≈ 34.416m
East Component = 300m * cos(90°)
= 300m * 1
= 300m
Total Displacement = sqrt(34.416^2 + 300^2)
= sqrt(1186.380096 + 90000)
≈ sqrt(91186.380096)
≈ 301.976m
Therefore, the boy is approximately 301.976 meters away from his starting point.
the x distance is 60sin35° + 300
the y distance is 60cos35°
the final distance is found using the distance formula as usual
or, it can also come from the law of cosines:
z^2 = x^2+y^2 - 2xy cos145°