Problem 175 - 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 are the coordinates of the child after 90 seconds?

Step-by-Step Solution

Answer

The coordinates of the child after 90 seconds are \((0,-1)\).

Step 1: Determine Carousel Revolutions

To find the number of revolutions the carousel makes, divide the total time in seconds by the time taken for one full revolution: \(90 \text{ seconds} \div 60 \text{ seconds/revolution} = 1.5 \text{ revolutions}\).

Step 2: Analyze Carousel's Rotation Pattern

Since the carousel completes one full revolution in 60 seconds, half a revolution (or 180 degrees counter-clockwise) will take an additional 30 seconds. This implies that after 1 complete revolution, the child moves from \((0,1)\) to halfway around the circle.

Step 3: Determine Final Position

After one full revolution (60 seconds), the child returns to the starting position \((0,1)\). For the remaining 30 seconds, the child moves another half-revolution (180 degrees). Since the carousel rotates counter-clockwise, a half-revolution from \((0,1)\) shifts the child's position to the exact opposite point on the circle's circumference, which is \((0,-1)\).

Key Concepts

Coordinate Geometry

To understand the child's position on the carousel, we can use the principles of coordinate geometry. The carousel moves in a circular path, and we can represent this on a coordinate plane. Anytime you're dealing with circular movements, you will think of coordinate geometry to map out those movements.Firstly, imagine a circle centered at the origin \((0,0)\) on a coordinate plane. The child's initial position is at \((0, 1)\), which is the northernmost point on the circle. As the carousel rotates, the child's position changes along the track of the circle's circumference.In coordinate geometry, every point on the circle can be expressed through its \(x\) and \(y\) coordinates. The child's movement around the circle adheres to the properties of a circle: constant distance from the center.

• The radius of the circle remains constant.
• The center of the circle is the carousel's axle, at \((0,0)\).

Coordinate geometry helps to visualize and calculate the exact movement path of an object on a circular trajectory.

Rotational Motion

Rotational motion pertains to objects moving in a circular path, around a central point. In this problem, the carousel represents rotational motion as it revolves around its center. Let's see how to analyze this.The carousel makes one full revolution in 60 seconds. Two important aspects of rotational motion to grasp are:

• Revolution: One complete circle around, which in this case takes 60 seconds.
• Angular displacement: The angle through which an object rotates during its motion. When we discuss a half revolution, this translates to \(180^{\circ}\) or \(\pi\) radians.

After 90 seconds, the carousel has completed 1.5 revolutions:- The first revolution returns the child to the starting point at \((0,1)\).- The additional half revolution moves the child to the diametrically opposite position at \((0,-1)\).Understanding rotational motion helps explain how the child moves from point to point on the circle's path within a given timeframe.

Trigonometric Functions

Trigonometry is a pivotal part of understanding rotations on a circle. It connects the rotation angle with the circle's radius and the position on the circle's perimeter.For circles, trigonometric functions such as sine and cosine become extremely handy to describe positions on the circle:

• Cosine corresponds to the x-coordinate, and sine to the y-coordinate, of a point on the circle.
• A full revolution uses angles measured in radians or degrees. For example, a 360-degree or \(2\pi\) radian rotation represents a complete circle.

The child starts at \((0,1)\), which can be interpreted using trigonometry:- At \(0^{\circ}\), the initial point is where sine equals 1 and cosine equals 0.- After a full revolution and then another half, the new angle is \(180^{\circ}\) from the start, flipping the sine value, resulting in the position \((0,-1)\).Trigonometric functions simplify the analysis of movements around circular paths, allowing us to determine positions quickly and accurately.