Problem 87 - Algebra and Trigonometry Real Mathematics, Real People

Question

The center of a Ferris wheel lies at the pole of the polar coordinate system, where the distances are in feet. Passengers enter a car at \((30,-\pi / 2) .\) It takes 45 seconds for the wheel to complete one clockwise revolution. (a) Write a polar equation that models the possible positions of a passenger car. (b) Passengers enter a car. Find and interpret their coordinates after 15 seconds of rotation. (c) Convert the point in part (b) to rectangular coordinates. Interpret the coordinates.

Step-by-Step Solution

Answer

The polar equation for the Ferris wheel is \( r(t) = 30, θ(t) = -2 \pi ( t / 45 ) - \pi / 2 \). After 15 seconds, the passengers' coordinates in polar form are \( (30 , -5\pi / 6) \), and in rectangular form, the coordinates are \( (-15\sqrt{3}, -15) \).

Step 1: Define the polar equation for the Ferris wheel

The radius is given as 30 feet and the rotation direction is clockwise. For a cycle of 45 seconds, a convenient model for this wheel gives the passenger's location at \(t\) seconds as: \(r( t ) = 30, θ( t ) = -2 \pi ( t / 45 ) - \pi / 2 \)

Step 2: Calculate the passenger's coordinates after 15 seconds

Plug \( t = 15 \) into \( r(t) \) and \( θ(t) \). This gives \( r(15) = 30 \) and \( θ(15) = -2 \pi ( 15 / 45 ) - \pi / 2 = -\pi/2 - \pi/3 = -5\pi / 6 \). Therefore, the coordinates of the passengers after 15 seconds in polar coordinates are \( (30 , -5\pi / 6) \).

Step 3: Conversion from polar to rectangular coordinates

We use formulas \( x = r\cos(θ) \) and \( y = r\sin(θ) \) to convert the polar coordinates to rectangular form. Plugging in the values, we get \( x = 30\cos(-5\pi / 6) = -15\sqrt{3} \) and \( y = 30\sin(-5\pi / 6) = -15\). Thus, the rectangular coordinates for the passengers after 15 seconds are \( (-15\sqrt{3}, -15) \). Please note, these coordinates mean that the passengers are 15 ft below the center of the Ferris wheel and approximately 26 ft to the left of the center.

Key Concepts

Ferris Wheel

A Ferris wheel is a popular amusement ride consisting of a vertical rotating wheel with passenger cars attached along its circumference. These cars move in a circular path as the wheel turns. The wheel's center typically serves as the origin in a polar coordinate system, making it easier to describe the positions of cars.
- The wheel rotates around its center, maintaining a constant radius. - Passengers board at specified coordinates when the wheel is in a stationary or controlled entry position.
In this exercise, a Ferris wheel has a radius of 30 feet and rotates once every 45 seconds, giving each passenger a smooth ride while tracing a perfect circular path.

Rectangular Coordinates

Rectangular coordinates are a way to represent points in a plane using two perpendicular lines, commonly defined as the x-axis and y-axis. In this system, a point's position is defined by two numbers: its horizontal and vertical distances from the origin, usually written as \((x, y)\). - Horizontal positions measure how far along the x-axis a point is. - Vertical positions measure how far along the y-axis a point is.

In our example, after converting from polar to rectangular coordinates, the position of the Ferris wheel car at 15 seconds is \((-15\sqrt{3}, -15)\). This means it is approximately 26 feet to the left of the wheel's center and 15 feet below it. Understanding this conversion is vital for interpreting positions within different types of coordinate systems.

Polar to Rectangular Conversion

The process of transforming points from polar to rectangular coordinates involves using trigonometric functions. Polar coordinates \((r, \theta)\) are defined by a radius from the origin and an angle from the positive x-axis.
To convert these into rectangular coordinates \((x, y)\), we use these formulas:

• \(x = r\cos(\theta)\)
• \(y = r\sin(\theta)\)

Applying this in our scenario for the coordinate \((30, -5\pi/6)\), we calculate

• \(x = 30\cos(-5\pi/6) = -15\sqrt{3}\)
• \(y = 30\sin(-5\pi/6) = -15\)

Converting coordinates allows us to switch from a perspective that involves angles to one that uses linear measurements, which can be more intuitive in certain contexts.

Clockwise Rotation

Clockwise rotation refers to the rotational direction similar to the hands of a clock. This is often used to describe motion in circular paths, such as that of a Ferris wheel.
- In polar coordinates, angles that increase positively are typically counter-clockwise. Thus, clockwise rotation involves decreasing the angle.
In this exercise, the Ferris wheel rotates clockwise, represented by negative angular changes as time progresses. Over 45 seconds, it completes a full rotation, which is effectively captured by the equation \(\theta(t) = -2\pi(t/45) - \pi/2\). Adding the initial \(-\pi/2\) aligns the starting point to where the passengers board, ensuring the calculations reflect this accurately and that their final position in coordinates makes sense based on the time elapsed and rotational direction.