You walk 51 m to the north, then turn 30° to your right and walk another 45 m. How far are you from where you originally started?
I suggest you draw the figure.
Note that if you draw the figure, you'll have a scalene triangle (a triangle with no sides equal). Two of its sides are given (51 and 45), and the angle between them is 180 - 30 = 150 degrees.
Since you have two given sides and an angle opposite to the required side length to find, we use the cosine law:
a^2 = b^2 + c^2 - 2bc * cos A
where
a = length we need to find
b & c = the given lengths (which are 51 and 45)
A = measure of angle opposite to side a (which is 150 degrees)
Substituting,
a^2 = 51^2 + 45^2 - 2(51)(45)*cos(150)
a^2 = 2601 + 2025 - (-3975)
a = sqrt(8601)
a = 92.74 m
Hope this helps :3
92.74 m
To find the distance from your original position, we can use the Pythagorean theorem.
1. Start by breaking down the motion into horizontal and vertical components.
- The 51 m northward walk can be represented as 51 m in the y-direction.
- The 45 m displacement after turning can be decomposed into horizontal and vertical components using trigonometry.
- The horizontal component = 45 m * cos(30°)
- The vertical component = 45 m * sin(30°)
2. Calculate the horizontal and vertical displacements:
- The horizontal displacement ≈ 45 m * cos(30°) ≈ 45 * 0.866 = 38.97 m
- The vertical displacement ≈ 45 m * sin(30°) ≈ 45 * 0.5 = 22.50 m
3. Use the Pythagorean theorem to find the total distance from the original starting point:
- Distance ≈ √(horizontal displacement^2 + vertical displacement^2)
- Distance ≈ √(38.97^2 + 22.50^2)
- Distance ≈ √(1521.2409 + 506.25)
- Distance ≈ √(2027.4909)
- Distance ≈ 45.05 m (rounded to two decimal places)
Therefore, you are approximately 45.05 meters away from where you originally started.
To solve this problem, we can use trigonometry and the concept of vectors.
Step 1: Draw a diagram representing the situation. Start with a point representing your original position and label it as "O". From there, draw a line segment of 51 meters pointing north. Label the endpoint of this line segment as "A". Then, measure an angle of 30 degrees from line OA (clockwise) and draw a line segment of 45 meters. Label the endpoint of this line segment as "B". Finally, draw a straight line connecting point O and point B. Label the point where the line intersects OB as "C".
Step 2: Now, we need to find the distance between point O and point C. To do this, we need to find the components of the displacement vector AC.
Step 3: The vertical component (ACy) can be calculated by multiplying the hypotenuse AC by the sine of the angle 30°. Since the hypotenuse AC is 45 meters, ACy = 45 * sin(30°).
Step 4: The horizontal component (ACx) can be calculated by multiplying the hypotenuse AC by the cosine of the angle 30°. Since the hypotenuse AC is 45 meters, ACx = 45 * cos(30°).
Step 5: Now, we need to calculate the horizontal displacement, which is the distance between the point A and point C. Since the vertical displacement is 51 meters, the horizontal displacement is ACx + 51.
Step 6: Finally, we can use the Pythagorean theorem to find the distance between the points O and C. The distance OC can be calculated as the square root of ((ACx + 51)^2 + ACy^2).
By following these steps and performing the calculations, we can find the answer to the question.