The Height Of A Seat On A Ferris Wheel Can Be Modeled As H ( T ) = 48 Sin ⁡ ( Π 30 T + 3 Π 2 ) + 53 H(t) = 48 \sin \left(\frac{\pi}{30} T + \frac{3 \pi}{2}\right) + 53 H ( T ) = 48 Sin ( 30 Π ​ T + 2 3 Π ​ ) + 53 , Where T T T Is Time In Seconds And H ( T H(t H ( T ] Is Height In Feet.How Far Off The Ground Is A Seat When

Introduction

Ferris wheels are a staple of amusement parks and fairs, providing a thrilling experience for riders of all ages. The height of a seat on a Ferris wheel can be modeled using a mathematical function, which can be used to determine the maximum height, minimum height, and other important characteristics of the ride. In this article, we will explore the mathematical model of a Ferris wheel seat and use it to determine how far off the ground a seat is at a given time.

The Mathematical Model

The height of a seat on a Ferris wheel can be modeled using the following function:

$$H(t) = 48 \sin \left(\frac{\pi}{30} t + \frac{3 \pi}{2}\right) + 53$$

where $t$ is time in seconds and $H(t)$ is height in feet. This function is a sinusoidal function, which means that it oscillates between a maximum and minimum value. The amplitude of the function is 48, which means that the height of the seat will oscillate between 53 - 48 = 5 feet and 53 + 48 = 101 feet.

Understanding the Function

To understand the function, let's break it down into its components. The function is a sinusoidal function, which means that it can be written in the form:

$$H(t) = A \sin(Bt + C) + D$$

where $A$ is the amplitude, $B$ is the frequency, $C$ is the phase shift, and $D$ is the vertical shift. In this case, $A = 48$, $B = \frac{\pi}{30}$, $C = \frac{3 \pi}{2}$, and $D = 53$.

Determining the Maximum and Minimum Heights

To determine the maximum and minimum heights of the seat, we need to find the values of $t$ that correspond to the maximum and minimum values of the function. The maximum value of the function occurs when the sine function is equal to 1, which happens when:

$$\frac{\pi}{30} t + \frac{3 \pi}{2} = \frac{\pi}{2}$$

Solving for $t$, we get:

$$t = 30$$

The minimum value of the function occurs when the sine function is equal to -1, which happens when:

$$\frac{\pi}{30} t + \frac{3 \pi}{2} = -\frac{\pi}{2}$$

Solving for $t$, we get:

$$t = 90$$

How Far Off the Ground is a Seat When?

To determine how far off the ground a seat is at a given time, we need to plug in the value of $t$ into the function. For example, if we want to know how far off the ground a seat is at $t = 10$ seconds, we can plug in this value into the function:

$$H(10) = 48 \sin \left(\frac{\pi}{30} (10) + \frac{3 \pi}{2}\right) + 53$$

Simplifying the expression, we get:

$$H(10) = 48 \sin \left(\frac{\pi}{3} + \frac{3 \pi}{2}\right) + 53$$

Using a calculator to evaluate the sine function, we get:

$$H(10) = 48 \sin \left(\frac{7 \pi}{6}\right) + 53$$

$$H(10) = 48 (-0.5) + 53$$

$$H(10) = -24 + 53$$

$$H(10) = 29$$

Therefore, at $t = 10$ seconds, the seat is 29 feet off the ground.

Conclusion

In this article, we explored the mathematical model of a Ferris wheel seat and used it to determine how far off the ground a seat is at a given time. We found that the height of the seat can be modeled using a sinusoidal function, which oscillates between a maximum and minimum value. We also determined the maximum and minimum heights of the seat and used the function to determine how far off the ground a seat is at a given time. This mathematical model can be used to design and optimize Ferris wheel rides, as well as to provide a more accurate and realistic experience for riders.

References

• [1] "Ferris Wheel Mathematics" by Math Is Fun
• [2] "The Mathematics of Ferris Wheels" by Wolfram MathWorld

Further Reading

• "The Mathematics of Amusement Park Rides" by Math Is Fun
• "The Physics of Ferris Wheels" by Physics Classroom

Mathematical Derivations

Derivation of the Maximum Height

To determine the maximum height of the seat, we need to find the value of $t$ that corresponds to the maximum value of the function. The maximum value of the function occurs when the sine function is equal to 1, which happens when:

$$\frac{\pi}{30} t + \frac{3 \pi}{2} = \frac{\pi}{2}$$

Solving for $t$, we get:

$$t = 30$$

Derivation of the Minimum Height

To determine the minimum height of the seat, we need to find the value of $t$ that corresponds to the minimum value of the function. The minimum value of the function occurs when the sine function is equal to -1, which happens when:

$$\frac{\pi}{30} t + \frac{3 \pi}{2} = -\frac{\pi}{2}$$

Solving for $t$, we get:

$$t = 90$$

Derivation of the Function

To derive the function, we can start with the general form of a sinusoidal function:

$$H(t) = A \sin(Bt + C) + D$$

where $A$ is the amplitude, $B$ is the frequency, $C$ is the phase shift, and $D$ is the vertical shift. In this case, $A = 48$, $B = \frac{\pi}{30}$, $C = \frac{3 \pi}{2}$, and $D = 53$.

Substituting these values into the general form of the function, we get:

$$H(t) = 48 \sin \left(\frac{\pi}{30} t + \frac{3 \pi}{2}\right) + 53$$

Introduction

In our previous article, we explored the mathematical model of a Ferris wheel seat and used it to determine how far off the ground a seat is at a given time. In this article, we will answer some of the most frequently asked questions about the height of a Ferris wheel seat.

Q: What is the maximum height of a Ferris wheel seat?

A: The maximum height of a Ferris wheel seat is 101 feet. This occurs when the sine function is equal to 1, which happens when:

$$\frac{\pi}{30} t + \frac{3 \pi}{2} = \frac{\pi}{2}$$

Solving for $t$, we get:

$$t = 30$$

Q: What is the minimum height of a Ferris wheel seat?

A: The minimum height of a Ferris wheel seat is 5 feet. This occurs when the sine function is equal to -1, which happens when:

$$\frac{\pi}{30} t + \frac{3 \pi}{2} = -\frac{\pi}{2}$$

Solving for $t$, we get:

$$t = 90$$

Q: How far off the ground is a seat when $t = 10$ seconds?

A: To determine how far off the ground a seat is at $t = 10$ seconds, we can plug in this value into the function:

$$H(10) = 48 \sin \left(\frac{\pi}{30} (10) + \frac{3 \pi}{2}\right) + 53$$

Simplifying the expression, we get:

$$H(10) = 48 \sin \left(\frac{\pi}{3} + \frac{3 \pi}{2}\right) + 53$$

Using a calculator to evaluate the sine function, we get:

$$H(10) = 48 (-0.5) + 53$$

$$H(10) = -24 + 53$$

$$H(10) = 29$$

Therefore, at $t = 10$ seconds, the seat is 29 feet off the ground.

Q: How far off the ground is a seat when $t = 60$ seconds?

A: To determine how far off the ground a seat is at $t = 60$ seconds, we can plug in this value into the function:

$$H(60) = 48 \sin \left(\frac{\pi}{30} (60) + \frac{3 \pi}{2}\right) + 53$$

Simplifying the expression, we get:

$$H(60) = 48 \sin \left(2 \pi + \frac{3 \pi}{2}\right) + 53$$

Using a calculator to evaluate the sine function, we get:

$$H(60) = 48 (0) + 53$$

$$H(60) = 53$$

Therefore, at $t = 60$ seconds, the seat is 53 feet off the ground.

Q: What is the period of the Ferris wheel seat height function?

A: The period of the Ferris wheel seat height function is 120 seconds. This is because the function repeats itself every 120 seconds, with the seat reaching its maximum height at $t = 30$ seconds and its minimum height at $t = 90$ seconds.

Q: What is the amplitude of the Ferris wheel seat height function?

A: The amplitude of the Ferris wheel seat height function is 48 feet. This is because the function oscillates between a maximum height of 101 feet and a minimum height of 5 feet, with the amplitude being the difference between these two values.

Conclusion

In this article, we answered some of the most frequently asked questions about the height of a Ferris wheel seat. We determined the maximum and minimum heights of the seat, as well as the period and amplitude of the function. We also used the function to determine how far off the ground a seat is at a given time. This mathematical model can be used to design and optimize Ferris wheel rides, as well as to provide a more accurate and realistic experience for riders.

References

• [1] "Ferris Wheel Mathematics" by Math Is Fun
• [2] "The Mathematics of Ferris Wheels" by Wolfram MathWorld

Further Reading

• "The Mathematics of Amusement Park Rides" by Math Is Fun
• "The Physics of Ferris Wheels" by Physics Classroom

Mathematical Derivations

Derivation of the Period

To determine the period of the Ferris wheel seat height function, we need to find the value of $t$ that corresponds to the period of the function. The period of the function is the time it takes for the function to repeat itself.

Let's start with the general form of a sinusoidal function:

$$H(t) = A \sin(Bt + C) + D$$

where $A$ is the amplitude, $B$ is the frequency, $C$ is the phase shift, and $D$ is the vertical shift. In this case, $A = 48$, $B = \frac{\pi}{30}$, $C = \frac{3 \pi}{2}$, and $D = 53$.

Substituting these values into the general form of the function, we get:

$$H(t) = 48 \sin \left(\frac{\pi}{30} t + \frac{3 \pi}{2}\right) + 53$$

To determine the period of the function, we need to find the value of $t$ that corresponds to the period of the function. The period of the function is the time it takes for the function to repeat itself.

Let's start by finding the value of $t$ that corresponds to the maximum height of the function. The maximum height of the function occurs when the sine function is equal to 1, which happens when:

$$\frac{\pi}{30} t + \frac{3 \pi}{2} = \frac{\pi}{2}$$

Solving for $t$, we get:

$$t = 30$$

Now, let's find the value of $t$ that corresponds to the minimum height of the function. The minimum height of the function occurs when the sine function is equal to -1, which happens when:

$$\frac{\pi}{30} t + \frac{3 \pi}{2} = -\frac{\pi}{2}$$

Solving for $t$, we get:

$$t = 90$$

The period of the function is the time it takes for the function to repeat itself. Since the function repeats itself every 120 seconds, the period of the function is 120 seconds.

Derivation of the Amplitude

To determine the amplitude of the Ferris wheel seat height function, we need to find the difference between the maximum and minimum heights of the function.

The maximum height of the function is 101 feet, and the minimum height of the function is 5 feet. The amplitude of the function is the difference between these two values:

$$A = 101 - 5$$

$$A = 96$$

However, we are given that the amplitude of the function is 48 feet. This means that the amplitude of the function is actually half of the difference between the maximum and minimum heights of the function.

Therefore, the amplitude of the Ferris wheel seat height function is 48 feet.