Let [tex]$t$[/tex] be the number of seconds that have elapsed since you began moving. It takes you 5 seconds to reach the top of the wheel (38 feet above the ground) and 20 seconds to make a complete revolution. The diameter of the wheel is 30 feet.
Work with your classmates to answer the following questions:
a. What is the lowest you go as the Ferris wheel turns?
b. Why is this number always greater than zero?
Write the equation of a sinusoid that describes this motion.
c. Predict your height above the ground when [tex]$t=3, t=8, t=15$[/tex], and [tex]$t=19$[/tex].
Answer (2)
The lowest height of the Ferris wheel is 8 feet, which is above zero because the center height (23 feet) is greater than the radius (15 feet). The sinusoidal equation for the height of the Ferris wheel is given by h e i g h t ( t ) = 15 cos ( 10 π ( t − 5 ) ) + 23 . The predicted heights at t = 3 , t = 8 , t = 15 , and t = 19 seconds are approximately 35.14, 31.82, 8, and 18.36 feet, respectively.
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Calculate the radius of the Ferris wheel: r a d i u s = 2 30 = 15 feet.
Determine the center height: ce n t er _ h e i g h t = 38 − 15 = 23 feet.
Calculate the lowest height: l o w es t _ h e i g h t = 23 − 15 = 8 feet.
Write the sinusoidal equation: h e i g h t ( t ) = 15 cos ( 10 π ( t − 5 ) ) + 23 , and predict heights at t = 3 , 8 , 15 , 19 .
The heights are approximately 35.14 , 31.82 , 8 , 18.36 feet, respectively.
The lowest point is always greater than zero because the center height (23 feet) is greater than the radius (15 feet).
h e i g h t ( t ) = 15 cos ( 10 π ( t − 5 ) ) + 23
Explanation
Problem Analysis Let's break down this Ferris wheel problem step by step! We'll start by finding the lowest point of the wheel, then explain why it's above ground. Next, we'll create an equation to model the Ferris wheel's motion, and finally, we'll use that equation to predict your height at different times.
Finding the Radius First, we need to determine the radius of the Ferris wheel. The diameter is given as 30 feet, so the radius is half of that: r a d i u s = 2 d iam e t er = 2 30 = 15 feet
Calculating the Center Height Next, we'll find the height of the center of the Ferris wheel. We know the highest point is 38 feet above the ground, and this is equal to the center height plus the radius. Therefore: ce n t er _ h e i g h t = t o p _ h e i g h t − r a d i u s = 38 − 15 = 23 feet
Determining the Lowest Height Now we can calculate the lowest point of the Ferris wheel. This is equal to the center height minus the radius: l o w es t _ h e i g h t = ce n t er _ h e i g h t − r a d i u s = 23 − 15 = 8 feet
Why the Lowest Height is Above Ground The lowest height is 8 feet above the ground. This is because the center of the Ferris wheel is 23 feet above the ground, which is more than the radius (15 feet). If the center height was less than the radius, the Ferris wheel would dip into the ground, which isn't possible in a real-world scenario!
Setting up the Sinusoidal Equation Now, let's create a sinusoidal equation to describe the Ferris wheel's motion. We'll use a cosine function because the Ferris wheel reaches its maximum height at t = 5 seconds. The general form of the equation is: h e i g h t ( t ) = A cos ( B ( t − C )) + D Where:
A is the amplitude (radius).
B = p er i o d 2 π
C is the horizontal shift.
D is the vertical shift (center height).
Determining the Parameters We already know:
Amplitude A = r a d i u s = 15 feet
Period = 20 seconds , so B = 20 2 π = 10 π ≈ 0.314
Horizontal shift C = 5 seconds (time to reach the top).
Vertical shift D = ce n t er _ h e i g h t = 23 feet
Writing the Sinusoidal Equation Plugging these values into the general equation, we get: h e i g h t ( t ) = 15 cos ( 10 π ( t − 5 ) ) + 23
Predicting Heights at Specific Times Finally, let's predict the height at t = 3 , 8 , 15 , and 19 seconds:
At t = 3 :
h e i g h t ( 3 ) = 15 cos ( 10 π ( 3 − 5 ) ) + 23 = 15 cos ( − 5 π ) + 23 ≈ 35.14 feet
At t = 8 :
h e i g h t ( 8 ) = 15 cos ( 10 π ( 8 − 5 ) ) + 23 = 15 cos ( 10 3 π ) + 23 ≈ 31.82 feet
At t = 15 :
h e i g h t ( 15 ) = 15 cos ( 10 π ( 15 − 5 ) ) + 23 = 15 cos ( π ) + 23 = 8 feet
At t = 19 :
h e i g h t ( 19 ) = 15 cos ( 10 π ( 19 − 5 ) ) + 23 = 15 cos ( 5 7 π ) + 23 ≈ 18.36 feet
Examples
Ferris wheels are a great example of periodic motion, which can be modeled using sinusoidal functions. Understanding these functions allows engineers to design and analyze rotating structures like Ferris wheels, ensuring safety and efficiency. Moreover, the same principles apply to other real-world scenarios, such as analyzing tidal patterns, the movement of pistons in an engine, or even the oscillations of a swing. By understanding the sinusoidal motion, we can predict the behavior of these systems over time.
