A boy walks 60m on a bearing of 035° and then 300m due east .How far is he from his starting point
We can solve this problem using trigonometry and vector addition.
- First, let's draw a diagram:
```
N
|
|
60m @ 035° | x
| /
| /
----*-----+------> E
|
|
|
```
The starting point is marked with a star (*), and we are given that the boy walks 60m on a bearing of 035°. This means he is walking towards the northeast, at an angle of 35° clockwise from north. The distance he walks is not important for now, as we only need to find the final distance from his starting point.
- Next, we can break down the second part of the journey into its horizontal and vertical components. The boy walks 300m due east, so his final position will be to the right (east) of his starting point, with no change in his vertical position:
```
N
|
|
| x -->
| /
| /
----*-----+------> E
|
|
|
```
- To find the distance between the two positions, we can add the two vectors together:
```
N
|
|
| x -->
| / \
| / \
----*-----+----------+------> E
| |
| |
| |
```
We can see that the two vectors form a right-angled triangle. To find the hypotenuse (the final distance from the starting point), we can use the Pythagorean theorem:
```
a^2 + b^2 = c^2
a = 60m
b = 300m
c = ?
c^2 = a^2 + b^2
c^2 = 60^2 + 300^2
c^2 = 36000 + 90000
c^2 = 126000
c = sqrt(126000)
c = 355.9m (rounded to one decimal place)
```
Therefore, the boy is 355.9m from his starting point.
335
I apologize for the mistake in my previous response. The correct answer is 335m. Here's a corrected explanation:
- The boy walks 60m on a bearing of 035°, which means he moves northeast. To find the horizontal and vertical components of this movement, we can use trigonometry:
```
cos(35°) = adjacent / hypotenuse
adjacent = cos(35°) * 60
adjacent = 49.4m
sin(35°) = opposite / hypotenuse
opposite = sin(35°) * 60
opposite = 34.4m
```
Therefore, the boy moves 49.4m to the east and 34.4m to the north.
- Next, the boy walks 300m due east. This adds to his eastward movement, but does not change his northward movement.
- To find the final distance between the boy's starting point and ending point, we can use the Pythagorean theorem:
```
distance^2 = east^2 + north^2
distance^2 = (60 + 300)^2 + 34.4^2
distance^2 = 36000 + 10816
distance^2 = 46816
distance = sqrt(46816)
distance = 216.4m
```
Therefore, the boy is 216.4m from his starting point. However, we need to add the 49.4m he walked to the east in the beginning. Therefore, the final answer is:
```
216.4m + 49.4m = 265.8m
```
The boy is 265.8m from his starting point. I apologize for any confusion caused by my previous response.
To find the distance from the starting point, we can use the Pythagorean theorem.
Step 1: Calculate the distance walked on a bearing of 035°.
We can break this distance into the horizontal and vertical components. The horizontal component can be calculated using the formula: Horizontal component = distance * cosine(bearing).
For the given distance of 60m and a bearing of 035°, the horizontal component can be calculated as:
Horizontal component = 60m * cosine(35°) = 60m * 0.819 = 49.14m.
The vertical component can be calculated using the formula: Vertical component = distance * sine(bearing).
For the given distance of 60m and a bearing of 035°, the vertical component can be calculated as:
Vertical component = 60m * sine(35°) = 60m * 0.574 = 34.44m.
So, the boy walks 49.14m horizontally and 34.44m vertically.
Step 2: Calculate the total horizontal distance after walking 300m due east.
Since the boy walks due east, there is no change in the horizontal distance, so the total horizontal distance remains 49.14m.
Step 3: Calculate the total vertical distance after walking 300m due east.
Since the boy walks due east and there is no vertical movement, the total vertical distance remains 34.44m.
Step 4: Use the Pythagorean theorem to calculate the distance from the starting point.
The distance from the starting point can be calculated using the formula: Distance = √(horizontal distance^2 + vertical distance^2).
Using the total horizontal distance of 49.14m and the total vertical distance of 34.44m:
Distance = √(49.14^2 + 34.44^2)
Distance = √(2412.5796 + 1186.4336)
Distance = √(3603.0132)
Distance = 60.02m (approximately).
Therefore, the boy is approximately 60.02 meters away from his starting point.
To find the distance from the starting point, we need to use the concept of vector addition. We can break down the movement into two components:
1. The first component is the boy walking 60m on a bearing of 035°. This can be represented as a vector in the north-east direction. We can find the horizontal and vertical components of this vector using trigonometry.
Horizontal Component = 60m * cos(35°)
Vertical Component = 60m * sin(35°)
2. The second component is the boy walking 300m due east. This can be represented as a vector in the east direction.
Now, let's add both vectors to find the resulting displacement from the starting point.
Horizontal Component + 300m = Resultant Eastward Component
Vertical Component + 0 = Resultant Northward Component
To find the magnitude of the resultant displacement, we can use the Pythagorean theorem:
Resultant Displacement = √((Resultant Eastward Component)^2 + (Resultant Northward Component)^2)
By substituting the values, we can calculate the distance.
Please note that the bearing is measured clockwise from the north direction, so a bearing of 035° means 35° to the east of north.