Problem 176 - Precalculus

Question

For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point \((0,1),\) that is, on the due north position. Assume the carousel revolves counter clockwise. What is the coordinates of the child after 125 seconds?

Step-by-Step Solution

Answer

The child's coordinates after 125 seconds are \((-\frac{\sqrt{3}}{2}, \frac{1}{2})\).

Step 1: Determine the Carousel's Revolution Time

The carousel makes one full revolution every 60 seconds. This means that every minute, the child completes a full circle and returns to the same starting point.

Step 2: Calculate Total Revolutions

Since the total time is 125 seconds, we need to determine how many full revolutions occur within this time. Divide the total time by the time for one revolution: \(\frac{125}{60} = \frac{5}{12} \) revolutions.

Step 3: Determine the Position in Partial Revolution

Since \(\frac{5}{12} \) of a revolution is not a complete turn, we need to find out the position on this partial path around the circle. Multiply \(\frac{5}{12} \) by 360 degrees to determine the angle: \(\frac{5}{12} \, \times \, 360 = 150 \) degrees counterclockwise from the starting position.

Step 4: Use Circular Rotation to Determine Coordinates

Starting from the north position \((0, 1)\), an angle of 150 degrees counterclockwise is equivalent to 180 degrees - 150 degrees = 30 degrees clockwise from east on standard position. Using cosine and sine for the 150 degrees position gives: \(x = \cos(150^\circ) = -\frac{\sqrt{3}}{2}\) and \(y = \sin(150^\circ) = \frac{1}{2}\). Hence, the coordinates are \((-\frac{\sqrt{3}}{2}, \frac{1}{2})\).

Key Concepts

Trigonometry in Circular Motion

When looking at circular motion, trigonometry is a vital tool to determine positions at various angles.
It's especially useful on platforms like carousels, where positions change in a circular path over time.
Angles in circular motion can be measured in degrees or radians.

• One full revolution around a circle is equal to 360 degrees or \(2\pi\) radians.
• In the given problem, time and rotation help us to determine the angular position using degrees.
  • For instance, after \(125\) seconds on our carousel, the child is at \(150\) degrees counterclockwise from the starting north position.

To find the exact position, we use trigonometric functions such as sine and cosine. These functions relate angles in a circle to coordinates on the Cartesian plane:

• Cosine determines the horizontal distance from the origin, giving us the \(x\)-coordinate.
• Sine determines the vertical distance, giving us the \(y\)-coordinate.

Thus, for an angle of \(150\) degrees, we find the coordinates to be:

• \(x = \cos(150^0) = -\frac{\sqrt{3}}{2}\)
• \(y = \sin(150^0) = \frac{1}{2}\)

Trigonometry allows us to convert rotational motion into precise positions on the 2D plane.

Coordinate Geometry of Circular Paths

Coordinate geometry provides a way to describe the positions on a circular path using the Cartesian coordinate system.
In our carousel example, the child's path is circular with the center of the circle at the origin (0,0), and the radius is 1 (since the initial position is at (0,1)).

• This setup is ideal for understanding how objects move in circles with a defined center and radius.
• A point on a circle can be identified using \((x, y) = (r \cdot \cos(\theta), r \cdot \sin(\theta))\), where \(\theta\) is the angle in radians and \(r\) is the radius.

For this exercise, \(\theta\) is the angle rotated over time, determined by the number of revolutions and their fractional parts:

• The angle \(150\) degrees corresponds to the child's position after partial rotation
• The transformations from angle to coordinates use the trigonometric values for sine and cosine at that specific angle.

By using relative angles in circular paths, coordinate geometry helps us precisely calculate positions like\((-\frac{\sqrt{3}}{2}, \frac{1}{2})\), encoding circular motion onto a simple grid.

Understanding Angular Displacement

Angular displacement is how far an object has moved around a circle in terms of angles.
It measures the change in the rotational position, differing from linear displacement that tracks straight-line movement.

• It's typically measured in degrees or radians.
• For our carousel problem, the complete circle equates to \(360\) degrees or \(2\pi\) radians.

The child in our scenario started at the north \((0,1)\) position & revolved \(\frac{5}{12}\) of a circle after 125 seconds:

• This is not a full rotation, so we calculate partial angular displacement by multiplying \(\frac{5}{12}\) by \(360\) degrees to get \(150\) degrees.

Using angular displacement, we understand how much of a circle path is covered:

• It helps us determine how far from the start an object has moved, simplifying position calculation.
• Knowing angular displacement lets us apply trigonometric functions to find specific points on the circle.

This concept is key to analyzing and predicting rotational motions, offering insights into precise positions after movements.