Problem 95 - Precalculus with Limits

Question

A Ferris wheel is built such that the height \( h \) (in feet) above ground of a seat on the wheel at time \( t \) (in minutes) can be modeled by \( h(t) = 53 + 50 \sin \left(\dfrac{\pi}{16} t - \dfrac{\pi}{2}\right) \). The wheel makes one revolution every \( 32 \) seconds. The ride begins when \( t = 0 \). (a) During the first \( 32 \) seconds of the ride, when will a person on the Ferris wheel be \( 53 \) feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts \( 160 \) seconds, how many times will a person be at the top of the ride, and at what times?

Step-by-Step Solution

Answer

a) The person will be 53 feet above ground at \( t = 8, 24 \) seconds.\n\nb) The person will be at the top of the Ferris wheel for the first time at \( t = 16 \) seconds. If the ride lasts 160 seconds, the person will be at the top 5 times at \( t = 16, 48, 80, 112, 144 \) seconds.

Step 1: Solving for Specific Height

For part a), the problem requires finding the time when the height, \( h \), reaches 53 feet. Solve the equation \( h(t) = 53 \) by setting the equation \( h(t) = 53 + 50 \sin \left(\dfrac{\pi}{16} t - \dfrac{\pi}{2}\right) \) equal to 53. After simplifying, the t-values within the range of one revolution, which is 32 seconds, can be calculated.

Step 2: Solving for the Peak Height

For part b), the peak of the Ferris wheel is at a height of 103 feet, the problem thus requires finding the time it takes to reach this height by solving \( h(t) = 103 \). The t-values can be calculated, taking into account: from the General Sinusoidal Equation \( y = A \sin \left(B(t-h)\right) + k \), \( B=\dfrac{\pi}{16} \), which means the period \( P = \dfrac{2\pi}{B} = 32 \) seconds.

Step 3: How Many times the Peak is Reached

To find out how many times and at what times the peak is reached during the 160 seconds ride, divide 160 by the period of the wheel which is 32 seconds. To find at what times the peak is reached multiply the period (32) by a whole number up to the number of times the peak is reached.

Key Concepts

Sinusoidal Functions

Sinusoidal functions are a type of periodic function that describe smooth and repetitive oscillations, much like the natural waves you might see in water or sound. These functions come in the form of sine or cosine functions, which are fundamental in trigonometry.
Sinusoidal functions can be expressed either as \( y = A \sin(B(t - C)) + D \) or \( y = A \cos(B(t - C)) + D \), where each variable plays a specific role:

• A is the amplitude, indicating maximum displacement from the midline.
• B affects the period of the function, showing how quickly the oscillations occur.
• C is the horizontal shift, moving the graph left or right.
• D is the vertical shift, which moves the function up or down.

Trigonometric functions like the sinusoidal functions are invaluable in modeling scenarios involving cycles and waves, such as sound waves, alternating currents, and in this example, the motion of a Ferris wheel.

Ferris Wheel Problem

The Ferris Wheel problem is a classic example of applying sinusoidal functions to real-world scenarios. In this problem, we determine the height of a Ferris wheel at different times, using a sinusoidal equation to model the circular motion.
The circular path of the Ferris wheel lends itself well to sine and cosine functions due to their cyclical nature. With the equation given for the Ferris wheel's height, \[ h(t) = 53 + 50 \sin \left(\dfrac{\pi}{16} t - \dfrac{\pi}{2}\right) \], we use it to figure out various parameters like when the Ferris wheel reaches a specific height.
In this context, each complete cycle of sine, within a given time, represents one full revolution of the wheel. Problems like this help students understand the versatility of trigonometric functions in solving practical problems.

Period of a Sinusoidal Function

The period of a sinusoidal function is the length of time it takes for the function to complete one full cycle and start to repeat itself. This period is crucial for problems involving cyclic events like the Ferris wheel rotation.
For the function \( h(t) = 53 + 50 \sin \left(\dfrac{\pi}{16} t - \dfrac{\pi}{2}\right) \), the period can be calculated using the formula \( P = \dfrac{2\pi}{B} \), where \( B \) is the coefficient of \( t \).
In this case, \( B = \dfrac{\pi}{16} \) so:\[ P = \dfrac{2\pi}{\dfrac{\pi}{16}} = 32 \text{ seconds} \] This means the Ferris wheel completes one full revolution every 32 seconds, revealing how trigonometric functions can effectively model time-based cyclic motions.

Height Modeling

Height modeling using sinusoidal functions allows us to predict the height of an object in cyclical motion, such as a Ferris wheel seat. This involves setting up an equation that accurately describes the up-and-down movement over time.
In our Ferris wheel example, the height function \( h(t) = 53 + 50 \sin \left(\dfrac{\pi}{16} t - \dfrac{\pi}{2}\right) \) tells us that:

• The vertical shift of 53 indicates the seating starts 53 feet above ground level.
• The amplitude of 50 means the maximum height variation from the central level is 50 feet.
• The negative phase shift of \(-\dfrac{\pi}{2}\) indicates the starting position of the function within its cycle.

Using this model, we can find specific moments when certain heights occur, like determining times when the seat is exactly 53 feet above ground or at the peak at 103 feet.

Revolution Time Calculation

Calculating the revolution time of a Ferris wheel involves understanding the period from the sinusoidal function. Here, each period signifies one complete turn or revolution of the wheel.
Given the function \( h(t) = 53 + 50 \sin \left(\dfrac{\pi}{16} t - \dfrac{\pi}{2}\right) \), the period calculated was 32 seconds, meaning it takes 32 seconds for the Ferris wheel to make one full revolution.
If a ride lasts for 160 seconds, we can calculate the total number of complete revolutions made by: \[ \text{Number of Revolutions} = \dfrac{\text{Total Time}}{\text{Period}} = \dfrac{160}{32} = 5 \text{ revolutions} \] This approach assists in determining how often particular events, such as reaching the top of the ride, will happen during the ride duration.