Problem 55 - Algebra 2

Question

Suppose the gondolas on the Navy Pier Ferris Wheel were numbered from 1 through 40 consecutively in a counterclockwise fashion. If you were sitting in gondola number 3 and the wheel were to rotate counterclockwise through \(\frac{47 \pi}{10}\) radians, which gondola used to be in the position that you are in now?

Step-by-Step Solution

Answer

Gondola 29 used to be in your position.

Step 1: Identify the Total Radians for One Full Rotation

The Ferris wheel is circular, so one full rotation is equivalent to the circumference in radians. Since a circle has a circumference of \(2\pi\), we will use \(2\pi\) radians for a full rotation.

Step 2: Calculate Equivalent Rotation in Radians

We know that the wheel rotates \(\frac{47\pi}{10}\) radians counterclockwise. To find out how many full rotations this is, we divide \(\frac{47\pi}{10}\) by \(2\pi\). The formula is: \(\text{Rotations} = \frac{47\pi/10}{2\pi} = \frac{47}{20} = 2.35\) rotations.

Step 3: Determine the Remaining Radian After Full Rotations

From \(2.35\) rotations, the whole number \(2\) represents complete rotations. We focus on the decimal part, \(0.35\) of a rotation, which is incomplete. To find the remaining radians, calculate \(0.35 \times 2\pi = 0.7 \pi\) radians.

Step 4: Calculate Number of Gondolas Moved After Last Full Rotation

Convert remaining radians (\(0.7 \pi\)) to gondola movements. Since the full wheel is \(40\) gondolas, each \(2\pi\) radians corresponds to \(40\) gondolas. Find the fraction of \(0.7\pi\) in terms of gondolas: \((0.7\pi / 2\pi) \times 40 = 0.35 \times 40 = 14\) gondolas.

Step 5: Determine Original Gondola Position After Rotate

Since you initially sit at gondola 3, moving \(14\) positions counterclockwise from position 3 lands you at the new gondola position. Calculate the original position using \(3 - 14 = -11\). Since there are \(40\) gondolas, wrap around using the modulus operation: \((-11 + 40) \mod 40 = 29\). Hence, gondola number \(29\) used to be in your position.

Key Concepts

Radian Measures

In trigonometry, radian measures provide a way to describe angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians are based on the concept that the length of an arc of a circle is directly proportional to the angle that subtends it at the center of the circle.
To understand radians, picture a circle with a radius of 1 unit, often called a unit circle. The circumference of this circle is calculated by the formula \(2\pi\) (since circumference = radius \(\times\) \(2\pi\)). Consequently, one complete rotation around the circle corresponds to an angle of \(2\pi\) radians.

A fraction of this circle also translates to a fraction of \(2\pi\) radians. For example, a quarter of a full circle (or 90 degrees) is equivalent to \(\frac{\pi}{2}\) radians. This measurement is crucial when dealing with rotational problems, such as determining travel distances along circular paths.

Circle Rotation

Circle rotation is a fundamental concept in geometry and trigonometry, addressing how points are repositioned around a central axis. Understanding circle rotation helps in visualizing motion along circular paths, such as with Ferris wheels or car tires.

Circle rotation can be clockwise or counterclockwise, with the direction impacting how we calculate and interpret rotational angles. In a counterclockwise direction, positive angles are used, while clockwise rotation involves negative angles. This distinction is key when solving problems about rotational positions.

• The rotation amount can be expressed in radians, which we previously discussed.
• The total angle for one full spin is \(2\pi\) radians.
• Part of a rotation can be measurable by fractions of \(2\pi\).

For example, our problem dictates that the wheel rotates \(\frac{47\pi}{10}\) radians counterclockwise, which we need to relate to gondola positions over multiple rotations.

Modulus Operation

The modulus operation is a mathematical tool that helps manage numbers within cyclic or repeating systems. It's crucial when dealing with circular objects, such as Ferris wheels, to calculate positions after a certain number of rotations.

In our problem, the modulus operation essentially "wraps" selections that exceed or go below certain defined limits back onto themselves. For example, determining which gondola ends up in a specific position involves these key steps:

• We first found the number of full rotations by dealing with radians and calculated the remaining partial rotation.
• After moving 14 gondolas counterclockwise, the calculation \(3 - 14\) led to \(-11\) because we reduced from position 3.
• To put this back in the range of our 40 gondolas, we use ((-11 + 40) mod 40), simplifying it to determine the original gondola position, which equals gondola 29.

This process demos how the modulus operation efficiently handles numbers that extend beyond existing sequences, making it immensely helpful in repetitive scenarios.