Problem 60 - Algebra and Trigonometry

Question

The seats of a Ferris wheel are 40 feet from the wheel's center. When you get on the ride, your seat is 5 feet above the ground. How far above the ground are you after rotating through an angle of \(765^{\circ} ?\) Round to the nearest foot.

Step-by-Step Solution

Answer

The seat is approximately 68 feet above the ground after the ferris wheel rotates through an angle of 765 degrees.

Step 1: Calculate the angle of rotation in the unit circle

Since one full rotation of the ferris wheel corresponds to 360 degrees, the equivalent position of 765 degrees rotation on the wheel is calculated as \(765 \mod 360 = 45\) degrees.

Step 2: Convert the angle to radians

To use the trigonometric function sin in the next step, we need to convert the angle from degrees to radians. The conversion factor is \(\pi/180\). Thus, we have \(45 * \pi/180 = \pi/4\) radians.

Step 3: Calculate the y-coordinate

The y-coordinate corresponds to the height above or below the center of the wheel, in feet. The y-coordinate is determined by \(y = r * \sin(\theta)\), where r is the radius of the ferris wheel (40 feet) and \(\theta\) is the angle of rotation in radians. Here, \(y = 40 * \sin(\pi/4) = 40 * \sqrt{2}/2 = 20*\sqrt{2}\) feet.

Step 4: Find the height above the ground

The height above the ground is the height above the center of the wheel plus the distance from the ground to the center of the wheel. So the height is \(20*\sqrt{2} + 40 = 40 + 20*\sqrt{2}\) feet. This should be rounded to the nearest foot.

Key Concepts

Understanding Angle of Rotation

In trigonometry and geometry, the concept of angle of rotation is crucial. It refers to the amount of rotation a point or line has undergone around a fixed point, often the origin of a coordinate system.
To better grasp this concept, think of a Ferris wheel. One point on the wheel moves in a circular path as the wheel turns. An angle of rotation tells us how far that point has moved along the circle.
Here's a simple breakdown:

• One full rotation around a circle is 360 degrees.
• So, if you rotate by 765 degrees, you will have completed two full rotations (720°) and rotated an additional 45 degrees.
• This remainder of 45 degrees places you at the same position as if you had rotated just that amount.

Recognizing these patterns is helpful not only in solving problems related to circular motion but also in understanding wider trigonometric concepts.

Simplifying Radian Conversion

Converting between degrees and radians is essential when working with trigonometric functions such as sine, cosine, and tangent. Radians offer a more natural measure for angles because they relate directly to the arc length of a circle.
Here's how to convert degrees to radians:

• Use the conversion formula: \[\text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]
• For example, 45 degrees can be converted to radians as \(45 \times \frac{\pi}{180} = \frac{\pi}{4}\).

The use of radians becomes particularly clear when plotting periodic functions like sine, which repeat every \(2\pi\) radians. Understanding radian conversion ensures that you accurately measure angles and effectively use trigonometric functions.

Exploring the Sin Function

The sine function, abbreviated as \(\sin\), is fundamental in trigonometry, often used to determine vertical distances on a coordinate plane.
When you see an equation like \(y = r \cdot \sin(\theta)\):

• \(y\) denotes the distance from the center (y-coordinate of a point on a circle).
• \(r\) stands for the radius of the circle, telling us how far the point is from the center.
• \(\theta\) is the angle in radians that represents the angle of rotation.

For example, using our Ferris wheel problem, with a radius \(r\) of 40 feet and an angle \(\theta\) of \(\frac{\pi}{4}\), you calculate the height using:\[y = 40 \cdot \sin\left(\frac{\pi}{4}\right) = 40 \cdot \frac{\sqrt{2}}{2} = 20\cdot \sqrt{2}\] This result gives you the vertical position of a point relative to the center of the circle, highlighting how trigonometric functions help find positions and distances on a circular path.