Find, correct to four decimal places, the length of the curve of intersection of the cylinder 
16x2 + y2 = 16
 and the plane 
x + y + z = 7

We are tasked with finding the length of the curve of intersection between the cylinder \(16x^2 + y^2 = 16\) and the plane \(x + y + z = 7\). Here is a step-by-step method to solve it:

### Step 1: Parametrize the Cylinder
The equation of the cylinder is \(16x^2 + y^2 = 16\). We can parametrize this curve by setting:
\[
x = \frac{1}{2} \cos t, \quad y = 4 \sin t
\]
where \(t\) is a parameter running over some interval.

### Step 2: Find z in Terms of t
From the equation of the plane \(x + y + z = 7\), we can solve for \(z\):
\[
z = 7 - (x + y)
\]
Substitute the parametrized expressions for \(x\) and \(y\):
\[
z = 7 - \left( \frac{1}{2} \cos t + 4 \sin t \right)
\]
Thus,
\[
z = 7 - \frac{1}{2} \cos t - 4 \sin t
\]

### Step 3: Parametrize the Curve of Intersection
The curve of intersection can now be represented parametrically as:
\[
\mathbf{r}(t) = \left( \frac{1}{2} \cos t, 4 \sin t, 7 - \frac{1}{2} \cos t - 4 \sin t \right)
\]
for some interval of \(t\).

### Step 4: Find the Derivative \(\mathbf{r}'(t)\)
Differentiate \(\mathbf{r}(t)\) with respect to \(t\):
\[
\mathbf{r}'(t) = \left( -\frac{1}{2} \sin t, 4 \cos t, \frac{1}{2} \sin t - 4 \cos t \right)
\]

### Step 5: Find the Magnitude of \(\mathbf{r}'(t)\)
The magnitude of \(\mathbf{r}'(t)\) is:
\[
|\mathbf{r}'(t)| = \sqrt{\left(-\frac{1}{2} \sin t\right)^2 + (4 \cos t)^2 + \left(\frac{1}{2} \sin t - 4 \cos t\right)^2}
\]
Simplifying this expression:
\[
|\mathbf{r}'(t)| = \sqrt{\frac{1}{4} \sin^2 t + 16 \cos^2 t + \left(\frac{1}{2} \sin t - 4 \cos t\right)^2}
\]
Expanding the square term:
\[
\left(\frac{1}{2} \sin t - 4 \cos t\right)^2 = \frac{1}{4} \sin^2 t - 4 \sin t \cos t + 16 \cos^2 t
\]
Thus, the magnitude becomes:
\[
|\mathbf{r}'(t)| = \sqrt{\frac{1}{4} \sin^2 t + 16 \cos^2 t + \frac{1}{4} \sin^2 t - 4 \sin t \cos t + 16 \cos^2 t}
\]
\[
|\mathbf{r}'(t)| = \sqrt{\frac{1}{2} \sin^2 t + 32 \cos^2 t - 4 \sin t \cos t}
\]

### Step 6: Integrate to Find the Length
The length of the curve is given by:
\[
L = \int_{0}^{2\pi} |\mathbf{r}'(t)| \, dt
\]
We now evaluate this integral either numerically or analytically to find the exact length. Let me compute that next.The length of the curve of intersection, correct to four decimal places, is \( 23.0839 \).

Would you like further details or explanations?

Here are 5 related questions you could explore:
1. How would the result change if the plane equation was different?
2. Can this method be applied to other types of cylinders?
3. What if the cylinder was elliptical instead of circular?
4. How do parametric equations help in solving curve length problems?
5. What is the geometric meaning of the intersection of a cylinder and a plane?

**Tip:** Parametrizing curves is a powerful technique for solving complex geometry problems, especially in three dimensions!

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 = 7.

Explore the detailed step-by-step solution to find the length of the curve of intersection between the cylinder 16x^2 + y^2 = 16 and the plane x + y + z = 7. Learn about parametric equations, vector calculus, and curve length calculation with a final answer accurate to four decimal places. Suitable for undergraduate students studying calculus and geometry.

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