Question

The helix of radius R, height h, and N complete turns has the parametrization

r(t) = (Rcos (2πNt/h), Rsin(2πNt/h), t.), 0 ≤ t ≤ h
Calculate the length s of a helix if R = 3, h = 19, and N = 4. (Use symbolic notation and fractions where needed.)

Answer 1

To calculate the length of the helix, use the formula for the arc length of a curve parametrized by a vector function. Substitute the given values into the parametrization and derivative to calculate the length.

Step-by-step explanation:

To calculate the length of the helix, we can use the formula for the arc length of a curve parametrized by a vector function. The formula is given by:

s = ∯ ||r'(t)|| dt, where r'(t) is the derivative of the vector function r(t).

In this case, the derivative of r(t) is given by:

r'(t) = (-2πNRsin(2πNt/h), 2πNRcos(2πNt/h), 1).

Substituting the values R = 3, h = 19, and N = 4 into the parametrization and the derivative, we can calculate the length s.