A highway curve forms a section of a circle. A car goes around the curve. Its dashboard compass shows that the car is initially heading due east. After it travels 620. m, it is heading 30.0° south of east. Find the radius of curvature of its path.
To find the radius of curvature of the car's path, we can use the concept of the angle subtended by an arc on the circumference of a circle.
Here's how we can solve it step by step:
Step 1: Draw a diagram representing the scenario. Label the position where the car starts as point A on the east side of the circle's center, and the final position after traveling 620. m as point B on the south side of the circle's center.
Step 2: Draw a line segment connecting points A and B. Label its length as the distance traveled, which is 620. m.
Step 3: Draw a line segment from the center of the circle to the midpoint of the line segment AB. Label that point as point O.
Step 4: Draw a perpendicular line from point O to the line segment AB. Label the point where the perpendicular line intersects AB as point C.
Step 5: Since the car initially heads due east and ends up heading 30.0° south of east, it traveled along the arc of a circle with a central angle of 30.0°.
Step 6: The line segment OC is the radius of the circle, which we need to find.
Step 7: According to the geometry of a circle, the central angle θ is related to the arc length s and the radius r by the formula: θ = s / r. We can rearrange this formula to solve for the radius: r = s / θ.
Step 8: Plug in the given values: s = 620. m and θ = 30.0°.
Step 9: Convert the angle from degrees to radians by multiplying it by (π / 180) to get: θ = (30.0°) * (π / 180) = 0.524 radians.
Step 10: Substitute the values of s and θ into the formula for the radius: r = 620. m / 0.524 = 1182.78 m.
Therefore, the radius of curvature of the car's path is approximately 1182.78 m.
To find the radius of curvature of the car's path, we can use the relationship between the radius and the change in direction of the car.
The change in direction can be calculated by finding the angle between the initial and final directions of the car's heading.
Given that the car initially heads due east and ends up heading 30.0° south of east, we can find the change in direction as follows:
Change in direction = 180° - 30.0° = 150.0°
Now, we can use this change in direction to find the radius of curvature using the following formula:
Radius of curvature = (Arc length) / (Change in direction)
The arc length is given as 620. m, so we can plug in the values and solve for the radius of curvature:
Radius of curvature = 620. m / 150.0°
Converting the angle from degrees to radians:
Radius of curvature = 620. m / (150.0° * (π/180))
Simplifying:
Radius of curvature = (620. m * π) / (150.0° * 180)
Calculating:
Radius of curvature ≈ 7.72 m
Therefore, the radius of curvature of the car's path is approximately 7.72 meters.