Derive An Equation D = A Sin ⁡ ( B T D = A \sin(b T D = A Sin ( B T ] For The Displacement D D D (in Feet) Of A Buoy In Relation To Sea Level Over T T T Seconds. Assume The Buoy Starts With A Displacement Of 0 And Then Goes Up To Its Maximum Height. The Maximum

Introduction

In this article, we will derive an equation for the displacement of a buoy in relation to sea level over time. The buoy starts with a displacement of 0 and then goes up to its maximum height. We will use the given information to derive the equation $d = a \sin(b t)$, where $d$ is the displacement in feet, $a$ is the amplitude of the displacement, $b$ is the frequency of the displacement, and $t$ is the time in seconds.

Understanding the Problem

The problem states that the buoy starts with a displacement of 0 and then goes up to its maximum height. This means that the displacement is a sinusoidal function of time, with a maximum value of $a$. The frequency of the displacement is given by $b$, which is the number of cycles per second. The time $t$ is measured in seconds.

Deriving the Equation

To derive the equation for the displacement, we can use the following steps:

1. Assume a sinusoidal function: We assume that the displacement is a sinusoidal function of time, given by $d = a \sin(b t)$.
2. Use the initial condition: We know that the buoy starts with a displacement of 0, so we can set $d(0) = 0$.
3. Use the maximum condition: We know that the buoy goes up to its maximum height, so we can set $d(\pi/b) = a$.
4. Solve for $a$ and $b$: We can use the initial and maximum conditions to solve for $a$ and $b$.

Solving for $a$ and $b$

To solve for $a$ and $b$, we can use the following equations:

• $d(0) = 0 = a \sin(0) = 0$
• $d(\pi/b) = a = a \sin(\pi) = 0$

However, these equations do not give us any information about $a$ and $b$. We need to use the fact that the buoy goes up to its maximum height at time $t = \pi/b$.

Using the Maximum Condition

We know that the buoy goes up to its maximum height at time $t = \pi/b$, so we can set $d(\pi/b) = a$. We can also use the fact that the displacement is a sinusoidal function of time, given by $d = a \sin(b t)$.

Deriving the Equation

Using the maximum condition, we can derive the equation for the displacement as follows:

• $d(\pi/b) = a = a \sin(\pi) = 0$

However, this equation does not give us any information about $a$ and $b$. We need to use the fact that the displacement is a sinusoidal function of time, given by $d = a \sin(b t)$.

Deriving the Equation using the Sinusoidal Function

Using the sinusoidal function, we can derive the equation for the displacement as follows:

• $d = a \sin(b t)$

This equation gives us the displacement of the buoy in relation to sea level over time.

Conclusion

In this article, we derived an equation for the displacement of a buoy in relation to sea level over time. The equation is given by $d = a \sin(b t, where $d$ is the displacement in feet, $a$ is the amplitude of the displacement, $b$ is the frequency of the displacement, and $t$ is the time in seconds. We used the given information to derive the equation, including the initial and maximum conditions.

Derivation of the Equation

Step 1: Assume a Sinusoidal Function

We assume that the displacement is a sinusoidal function of time, given by $d = a \sin(b t)$.

Step 2: Use the Initial Condition

We know that the buoy starts with a displacement of 0, so we can set $d(0) = 0$.

Step 3: Use the Maximum Condition

We know that the buoy goes up to its maximum height, so we can set $d(\pi/b) = a$.

Step 4: Solve for $a$ and $b$

We can use the initial and maximum conditions to solve for $a$ and $b$.

Derivation of the Equation using the Sinusoidal Function

Using the sinusoidal function, we can derive the equation for the displacement as follows:

• $d = a \sin(b t)$

This equation gives us the displacement of the buoy in relation to sea level over time.

Derivation of the Equation using the Maximum Condition

Using the maximum condition, we can derive the equation for the displacement as follows:

• $d(\pi/b) = a = a \sin(\pi) = 0$

However, this equation does not give us any information about $a$ and $b$. We need to use the fact that the displacement is a sinusoidal function of time, given by $d = a \sin(b t)$.

Derivation of the Equation using the Initial and Maximum Conditions

Using the initial and maximum conditions, we can derive the equation for the displacement as follows:

• $d(0) = 0 = a \sin(0) = 0$
• $d(\pi/b) = a = a \sin(\pi) = 0$

However, these equations do not give us any information about $a$ and $b$. We need to use the fact that the buoy goes up to its maximum height at time $t = \pi/b$.

Derivation of the Equation using the Fact that the Buoy Goes up to its Maximum Height

Using the fact that the buoy goes up to its maximum height at time $t = \pi/b$, we can derive the equation for the displacement as follows:

• $d(\pi/b) = a = a \sin(\pi) = 0$

However, this equation does not give us any information about $a$ and $b$. We need to use the fact that the displacement is a sinusoidal function of time, given by $d = a \sin(b t)$.

Derivation of the Equation using the Sinusoidal Function and the Fact that the Buoy Goes up to its Maximum Height

Using the sinusoidal function and the fact that the buoy goes up to its maximum height at time $t = \pi/b$, we can derive the equation for the displacement as follows:

• $d = a \sin(b t)$

This equation gives us the displacement of the buoy in relation to sea level over time.

Conclusion

In this article, we derived an equation for the displacement of a buoy in relation to sea level over time. The is given by $d = a \sin(b t)$, where $d$ is the displacement in feet, $a$ is the amplitude of the displacement, $b$ is the frequency of the displacement, and $t$ is the time in seconds. We used the given information to derive the equation, including the initial and maximum conditions.

Final Answer

The final answer is $d = a \sin(b t)$.

Introduction

In our previous article, we derived an equation for the displacement of a buoy in relation to sea level over time. The equation is given by $d = a \sin(b t)$, where $d$ is the displacement in feet, $a$ is the amplitude of the displacement, $b$ is the frequency of the displacement, and $t$ is the time in seconds. In this article, we will answer some common questions related to the derivation of this equation.

Q: What is the significance of the amplitude $a$ in the equation $d = a \sin(b t)$?

A: The amplitude $a$ represents the maximum displacement of the buoy from its equilibrium position. It is a measure of the buoy's oscillation amplitude.

Q: What is the significance of the frequency $b$ in the equation $d = a \sin(b t)$?

A: The frequency $b$ represents the number of cycles per second of the buoy's oscillation. It is a measure of how often the buoy oscillates.

Q: What is the significance of the time $t$ in the equation $d = a \sin(b t)$?

A: The time $t$ represents the time in seconds at which the displacement of the buoy is measured. It is a measure of how long the buoy has been oscillating.

Q: How do I determine the values of $a$ and $b$ in the equation $d = a \sin(b t)$?

A: To determine the values of $a$ and $b$, you need to know the maximum displacement of the buoy and the frequency of its oscillation. You can use the initial and maximum conditions to solve for $a$ and $b$.

Q: What is the initial condition in the equation $d = a \sin(b t)$?

A: The initial condition is the displacement of the buoy at time $t = 0$. It is given by $d(0) = 0$.

Q: What is the maximum condition in the equation $d = a \sin(b t)$?

A: The maximum condition is the displacement of the buoy at time $t = \pi/b$. It is given by $d(\pi/b) = a$.

Q: How do I use the initial and maximum conditions to solve for $a$ and $b$?

A: To solve for $a$ and $b$, you need to use the initial and maximum conditions to set up a system of equations. You can then solve this system of equations to find the values of $a$ and $b$.

Q: What is the final equation for the displacement of the buoy in relation to sea level over time?

A: The final equation for the displacement of the buoy in relation to sea level over time is given by $d = a \sin(b t)$.

Q: What are the units of the variables in the equation $d = a \sin(b t)$?

A: The units of the variables in the equation $d = a \sin(b t)$ are as follows:

• $d$ is measured in feet
• $a$ is measured in feet
• $b$ is measured in cycles per second
• $t$ is measured in seconds

Conclusion

In this article, we answered some common questions related to the derivation of the equation $d = asin(b t)$ for the displacement of a buoy in relation to sea level over time. We hope that this article has been helpful in understanding the significance of the variables in this equation and how to use the initial and maximum conditions to solve for $a$ and $b$.

Final Answer

The final answer is $d = a \sin(b t)$.