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Part 1
A circle of radius a that is centered at​ (A,0) is revolved about the​ y-axis to create a torus​ (assume aless than​A). When current flows through a copper wire that is wrapped around this​ torus, a magnetic field is created and the strength of this field depends on the amount of copper wire used. If the wire is wrapped evenly around the torus a total of k​ times, the shape of the wire is modeled by the following function. Determine the amount of copper required if Aequals4 ​in, aequals2 ​in, and kequals60​, using 0less than or equalstless than or equals2pi.
Bold r left parenthesis t right parenthesisequalsleft angle left parenthesis Upper A plus a cosine kt right parenthesis cosine t comma left parenthesis Upper A plus a cosine kt right parenthesis sine t comma a sine kt right angle


The problem asks to determine the amount of copper wire required to wrap around the torus given the parametric function modeling the wire's path. The equation provided for the position of the wire in three-dimensional space is:

\[
\mathbf{r}(t) = \langle (A + a \cos(kt)) \cos(t), \, (A + a \cos(kt)) \sin(t), \, a \sin(kt) \rangle
\]
where:
- \( A = 4 \) inches (the distance from the center of the torus to the center of the tube),
- \( a = 2 \) inches (the radius of the tube),
- \( k = 60 \) (the number of windings),
- \( 0 \leq t \leq 2\pi \) (the interval for the parameter \( t \)).

### Step 1: Compute the Arc Length Formula
To find the total length of the wire, we need to calculate the arc length of the curve defined by \( \mathbf{r}(t) \). The formula for the arc length \( L \) of a parametric curve \( \mathbf{r}(t) \) over an interval \( [t_1, t_2] \) is:

\[
L = \int_{t_1}^{t_2} \|\mathbf{r}'(t)\| \, dt
\]
where \( \mathbf{r}'(t) \) is the derivative of the parametric vector function \( \mathbf{r}(t) \), and \( \|\mathbf{r}'(t)\| \) is the magnitude of that derivative.

### Step 2: Differentiate the Parametric Function
To calculate \( \mathbf{r}'(t) \), we'll differentiate each component of \( \mathbf{r}(t) \):

1. First component: \( (A + a \cos(kt)) \cos(t) \)
   \[
   \frac{d}{dt} \left[ (A + a \cos(kt)) \cos(t) \right] = - (A + a \cos(kt)) \sin(t) - ak \sin(kt) \cos(t)
   \]

2. Second component: \( (A + a \cos(kt)) \sin(t) \)
   \[
   \frac{d}{dt} \left[ (A + a \cos(kt)) \sin(t) \right] = (A + a \cos(kt)) \cos(t) - ak \sin(kt) \sin(t)
   \]

3. Third component: \( a \sin(kt) \)
   \[
   \frac{d}{dt} \left[ a \sin(kt) \right] = ak \cos(kt)
   \]

### Step 3: Calculate the Magnitude of the Derivative
The magnitude of \( \mathbf{r}'(t) \) is given by:

\[
\|\mathbf{r}'(t)\| = \sqrt{\left( -(A + a \cos(kt)) \sin(t) - ak \sin(kt) \cos(t) \right)^2 + \left( (A + a \cos(kt)) \cos(t) - ak \sin(kt) \sin(t) \right)^2 + \left( ak \cos(kt) \right)^2}
\]

This expression can be simplified, but it involves considerable algebra. To avoid manual computation errors, we can compute this integral using numerical techniques.

### Step 4: Set Up the Arc Length Integral
Once we have the magnitude \( \|\mathbf{r}'(t)\| \), the total length of the wire (which corresponds to the amount of copper needed) is:

\[
L = \int_0^{2\pi} \|\mathbf{r}'(t)\| \, dt
\]

Let me calculate this integral for you to determine the required length of copper wire.The symbolic computation of the arc length took longer than expected due to the complexity of the expression. To resolve this, we can instead evaluate the integral numerically, which will be more efficient. I'll proceed with that method to find the total length of copper wire needed.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, you can attempt numerical integration using a calculator or other software to find the arc length, or I can guide you through the process step-by-step.

Do you have any other questions, or would you like further details on this problem? 

Here are five related questions to expand on this topic:
1. How is the arc length of a parametric curve generally calculated?
2. What does the parameter \( k \) represent in this context?
3. How would increasing the radius \( a \) affect the total length of the wire?
4. How would the calculation change if the wire was wrapped unevenly around the torus?
5. How does the choice of parametric equations model the wire's path?

**Tip:** When computing arc lengths, using numerical integration methods like Simpson's rule or built-in software functions can save time and reduce errors for complex curves.

A circle of radius a that is centered at (A,0) is revolved about the y-axis to create a torus (assume a < A). When current flows through a copper wire that is wrapped around this torus, a magnetic field is created and the strength of this field depends on the amount of copper wire used. If the wire is wrapped evenly around the torus a total of k times, the shape of the wire is modeled by the following function. Determine the amount of copper required if A = 4 in, a = 2 in, and k = 60, using 0 ≤ t ≤ 2π. r(t) = ⟨(A + a cos(kt)) cos(t), (A + a cos(kt)) sin(t), a sin(kt)⟩

Determine the amount of copper wire needed to wrap around a torus with parameters A = 4 in, a = 2 in, and k = 60 windings. This problem involves calculating the arc length of a parametric curve, using concepts of differentiation and numerical integration.

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