Final answer:
To find the length of the curve of intersection between the cylinder and the plane, you need to parameterize the curve using a parameter t and then use numerical methods to approximate the result.
Step-by-step explanation:
To find the length of the curve of intersection between the cylinder and the plane, we need to parameterize the curve using a parameter t. Let's express x and y in terms of t first. From the equation of the cylinder, we have 16x^2y^2 = 16. Dividing both sides by 16, we obtain x^2y^2 = 1. Taking the square root of both sides, we get xy = ±1.
Now, substituting x = t and y = 1/t into the equation of the plane, we have t(1/t)z = 14, which simplifies to z = 14. Therefore, the parameterization of the curve is x = t, y = 1/t, and z = 14.
The length of the curve can be found using the arc length formula: ∫ sqrt((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2) dt. Substituting the parameterization, we have ∫ sqrt(1 + (-1/t^2)^2 + 0) dt. Simplifying, we get ∫ sqrt(1 + 1/t^4) dt.
Since this integral does not have an elementary antiderivative, we need to use numerical methods to approximate the result. Using a numerical integration method, such as the trapezoidal rule or Simpson's rule, we can compute an approximate value of the integral to find the length of the curve of intersection.