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We are tasked with finding the length of the curve of intersection between the cylinder \(16x^2 + y^2 = 16\) and the plane \(x + y + z = 3\). To do this, we will:

### Step 1: Parametrize the curve of intersection

#### Cylinder equation:
The equation of the cylinder is:
\[
16x^2 + y^2 = 16
\]
We can rewrite this as:
\[
x^2 + \frac{y^2}{16} = 1
\]
This is the equation of an ellipse in the \(xy\)-plane, with \(x = 4\cos t\) and \(y = 4\sin t\), where \(t\) is a parameter.

#### Plane equation:
The equation of the plane is:
\[
x + y + z = 3
\]
Substitute \(x = 4\cos t\) and \(y = 4\sin t\) into the plane equation to express \(z\):
\[
4\cos t + 4\sin t + z = 3
\]
\[
z = 3 - 4\cos t - 4\sin t
\]

Thus, the parametric equations for the curve of intersection are:
\[
x = 4\cos t, \quad y = 4\sin t, \quad z = 3 - 4\cos t - 4\sin t
\]

### Step 2: Calculate the length of the curve

The formula for the length of a parametric curve \((x(t), y(t), z(t))\) is given by:
\[
L = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 + \left( \frac{dz}{dt} \right)^2} \, dt
\]
We will calculate the derivatives:
- \( \frac{dx}{dt} = -4\sin t \)
- \( \frac{dy}{dt} = 4\cos t \)
- \( \frac{dz}{dt} = 4\sin t + 4\cos t \)

Now substitute into the length formula:
\[
L = \int_0^{2\pi} \sqrt{(-4\sin t)^2 + (4\cos t)^2 + (4\sin t + 4\cos t)^2} \, dt
\]

Let me compute this next.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you through the manual steps to solve it yourself, or try a different method?

Here are some related questions you could consider:
1. How would the length of the curve change if the equation of the cylinder were different?
2. What is the effect of changing the plane equation on the shape of the curve of intersection?
3. Can we generalize this method to find the intersection of more complex surfaces?
4. How do parametric equations help in calculating curve lengths?
5. What are some common applications of finding the length of curves in physics or engineering?

**Tip**: When working with parametric curves, always ensure your parameter bounds correspond to the full extent of the curve for accurate calculations.

Find, correct to four decimal places, the length of the curve of intersection of the cylinder 16x^2 + y^2 = 16 and the plane x + y + z = 3.

Find the length of the curve of intersection between the cylinder 16x^2 + y^2 = 16 and the plane x + y + z = 3, using parametric equations and curve length formula. This multivariable calculus problem is suitable for college-level calculus students.

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