Problem 177 - 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. When will the child have coordinates \((0.707,-0.707)\) if the ride lasts 6 minutes? (There are multiple answers.)

Step-by-Step Solution

Answer

The child reaches \((0.707, -0.707)\) at minutes 0.875, 1.875, 2.875, 3.875, 4.875, and 5.875.

Step 1: Identify the Carousel's Motion

The carousel moves in a circular motion, and takes one minute to complete a full revolution. The initial point is \((0,1)\). This means the carousel starts at the top of the circle and revolves counter-clockwise.

Step 2: Understand the Target Point

The target coordinates are \((0.707, -0.707)\). This corresponds to a point on the unit circle in the fourth quadrant, since the x-coordinate is positive and the y-coordinate is negative. The angle associated with this point relative to the positive x-axis is \(\frac{7\pi}{4}\) or equivalently \(-\frac{\pi}{4}\).

Step 3: Convert the Angle to Time

Since the carousel takes one minute for a full revolution (i.e., \(2\pi\) radians), an angle of \(-\frac{\pi}{4}\) or \(\frac{7\pi}{4}\) corresponds to part of this minute. Calculate the fraction of a minute for this angle: \[\text{Time} = \frac{-\frac{\pi}{4}}{2\pi} \times 1 \text{ minute} = -\frac{1}{8} \text{ minute}\] which is equivalent to \(\frac{7}{8}\) of a minute by adding 1 minute.

Step 4: Account for Multiple Revolutions

The ride lasts for 6 minutes, allowing for multiple revolutions. Every full revolution adds multiples of 1 to the time. Thus, potential times when the child reaches the point \((0.707, -0.707)\) are \(\frac{7}{8} + n\) minutes, where \(n\) is the number of full revolutions. Solving for \(n\), the possible times within the 6-minute ride are:\[\frac{7}{8}, \frac{7}{8} + 1, \frac{7}{8} + 2, \frac{7}{8} + 3, \frac{7}{8} + 4, \frac{7}{8} + 5\]

Step 5: Calculate Specific Times

The specific times the child reaches the point \((0.707, -0.707)\) are: \[\frac{7}{8} \approx 0.875,\]\[\frac{15}{8} = 1.875,\]\[\frac{23}{8} = 2.875,\]\[\frac{31}{8} = 3.875,\]\[\frac{39}{8} = 4.875,\]\[\frac{47}{8} = 5.875\]These represent the minutes within the ride that the child is at those coordinates.

Key Concepts

Unit Circle

The unit circle is an essential concept in understanding circular motion. It is a circle with a radius of 1 centered at the origin of a coordinate plane. When working with problems involving angles and coordinate points, the unit circle provides valuable insight. Every point on the unit circle corresponds to an angle from the positive x-axis, measured in either degrees or radians.
In this exercise, the carousel starts at the point (0,1), which is at the top of the unit circle, known as the "due north position." As the carousel revolves counter-clockwise, different angles map to different points on the circle. Knowing this helps us understand how motion around the circle translates into angle measures.
The target point of (0.707, -0.707) is in the fourth quadrant, indicating that both the x and y movements are significant but in different directions. Understanding such coordinates can be easily tackled using the concept of the unit circle.

Radians

Radians are a way of measuring angles based on the radius of a circle. One radian is the angle created when the radius length is wrapped along the circumference. The full circle has an angle of 2\(\pi\) radians, which equals 360 degrees. Thus, radians provide a more natural and mathematically convenient way to express angles, especially in circular motion, because they relate directly to the circle’s dimensions.
In the carousel scenario, we often convert positions on the unit circle into radians. The target point (0.707, -0.707) represents an angle of \(\frac{7\pi}{4}\) radians, or equivalently, \(-\frac{\pi}{4}\) radians. This conversion helps us calculate the time needed for a child to reach a specific point along the ride.
Using radians simplifies finding how far along the circle a particular position is, and subsequently, how much time it represents in a scenario involving uniform circular motion like this carousel ride.

Revolution

A revolution refers to a complete turn around a circle. In circular motion, such as the carousel ride in this exercise, one revolution corresponds to a 360-degree turn or an angle of 2\(\pi\) radians. Understanding revolutions is crucial when dealing with periodic motions, as it helps track positions over time in terms of complete cycles.
In the problem, the carousel completes one revolution per minute. This means each full turn takes one minute, and the child circles back to the start point. Since the ride lasts for 6 minutes, the child makes multiple revolutions, leading to the possibility of revisiting points multiple times.
By calculating the points reached during each revolution, you can identify the exact times when the child arrives at specific coordinates like (0.707, -0.707). Given our unit circle understanding, revolutions help us systematically predict the points visited during each cycle and ensure all possible occurrences within the ride duration are accounted for.