Interpreting the Sine Function

Oct 13, 2025 by leolavanderia.com.br 148 views

Introduction

In this problem, we are given a scenario where two friends are sitting on a seesaw at the park. The height of the left seat above the ground is modeled by a sine function, where $t$ represents the time in seconds since the friends started sitting on the seesaw. Our goal is to determine the correct answer based on the given information.

The Sine Function Model

The height, $h(t)$ , of the left seat above the ground is modeled by a sine function. This means that the height of the seat varies sinusoidally with time. The sine function is given by:

h(t) = a \sin (b t + c) + d

where $a$ , $b$ , $c$ , and $d$ are constants that determine the shape and position of the sine wave.

Interpreting the Sine Function

To understand the behavior of the seesaw, we need to interpret the sine function. The amplitude of the sine wave, $a$ , represents the maximum height of the seat above the ground. The period of the sine wave, $\frac{2 \pi}{b}$ , represents the time it takes for the seat to complete one full cycle of motion. The phase shift, $-\frac{c}{b}$ , represents the initial position of the seat at time $t = 0$ . The vertical shift, $d$ , represents the average height of the seat above the ground.

Determining the Correct Answer

To determine the correct answer, we need to analyze the given information and use our understanding of the sine function model. However, since the problem does not provide specific values for the constants $a$ , $b$ , $c$ , and $d$ , we cannot determine a specific numerical answer. Instead, we can provide a general solution based on the properties of the sine function.

General Solution

Based on the properties of the sine function, we can conclude that the height of the left seat above the ground varies sinusoidally with time. The maximum height of the seat is given by $a$ , and the minimum height is given by $-a$ . The seat completes one full cycle of motion in a time period of $\frac{2 \pi}{b}$ seconds. The initial position of the seat at time $t = 0$ is given by $-\frac{c}{b}$ , and the average height of the seat above the ground is given by $d$ .

Conclusion

In conclusion, the problem provides a scenario where two friends are sitting on a seesaw at the park, and the height of the left seat above the ground is modeled by a sine function. We have analyzed the sine function model and determined the general solution based on the properties of the sine function. However, since the problem does not provide specific values for the constants $a$ , $b$ , $c$ , and $d$ , we cannot determine a specific numerical answer.

Key Takeaways

The height of the left seat above the ground varies sinusoidally with time.
The maximum height of the seat is given by $a$ .
The minimum height of the seat is given by $-a$ .
The seat completes one full cycle of motion in a time period of $\frac{2 \pi}{b}$ seconds.
The initial position of the seat at time $t = 0$ is given by $-\frac{c}{b}$ .
The average height of the seat above the ground is given by $d$ .

Further Discussion

Frequently Asked Questions

Q: What is the sine function model, and how does it relate to the seesaw? A: The sine function model is a mathematical representation of the height of the left seat above the ground as a function of time. It describes how the height of the seat varies sinusoidally with time, taking into account the maximum height, minimum height, period, phase shift, and vertical shift.

Q: What are the key components of the sine function model? A: The key components of the sine function model are:

Amplitude (a): represents the maximum height of the seat above the ground
Period (b): represents the time it takes for the seat to complete one full cycle of motion
Phase shift (c): represents the initial position of the seat at time t = 0
Vertical shift (d): represents the average height of the seat above the ground

Q: How does the sine function model describe the motion of the seesaw? A: The sine function model describes the motion of the seesaw as a sinusoidal function of time, where the height of the seat varies in a periodic manner. The model takes into account the maximum and minimum heights of the seat, as well as the time it takes for the seat to complete one full cycle of motion.

Q: What are the implications of the sine function model for the seesaw? A: The sine function model has several implications for the seesaw, including:

The seat will reach its maximum height at regular intervals
The seat will reach its minimum height at regular intervals
The seat will complete one full cycle of motion in a time period of 2π/b seconds
The initial position of the seat at time t = 0 will be given by -c/b

Q: How can the sine function model be used in real-world applications? A: The sine function model can be used in a variety of real-world applications, including:

Modeling the motion of pendulums and other oscillating systems
Describing the behavior of waves and other periodic phenomena
Analyzing the motion of objects in physics and engineering

Q: What are some common mistakes to avoid when using the sine function model? A: Some common mistakes to avoid when using the sine function model include:

Failing to account for the amplitude, period, phase shift, and vertical shift of the sine function
Using the wrong values for the constants a, b, c, and d
Failing to consider the implications of the sine function model for the seesaw

Q: How can the sine function model be extended to more complex systems? A: The sine function model can be extended to more complex systems by incorporating additional variables and parameters. For example, the model can be modified to include damping, friction, or other external forces that affect the motion of the seesaw.

Conclusion

In conclusion, the sine function model provides a powerful tool for understanding the motion of the seesaw. By analyzing the key components of the model and considering the implications of the model for the seesaw, we can gain a deeper understanding of the behavior of the system. Whether you are a student, teacher, or researcher, the sine function model is an essential tool for anyone interested in mathematics, physics, and engineering.