Arc Length with Vector Functions — Question 9

Question 9 – Solution

Strategy. The model has constant speed; after integration, solve the design equation by squaring positive quantities.

Step 1: General formula $$\mathbf r'(t)=\left\langle -R\sin t,R\cos t,\frac{H}{2\pi N}\right\rangle,$$ so $$\|\mathbf r'(t)\|=\sqrt{R^2+\left(\frac{H}{2\pi N}\right)^2}.$$ Multiplying this constant speed by the parameter length $2\pi N$ gives $$\boxed{L=\sqrt{(2\pi NR)^2+H^2}}.$$

Step 2: Apply the data With $R=3$, $H=24$, and $L=30\pi$, $$(30\pi)^2=(6\pi N)^2+24^2.$$ Therefore $$\begin{align*} 36\pi^2N^2&=900\pi^2-576,\\ N^2&=25-\frac{16}{\pi^2}. \end{align*}$$ Since $N>0$, $$\boxed{N=\sqrt{25-\frac{16}{\pi^2}}}.$$

Step 3: Feasibility Numerically, $N\approx 4.835$, which is not an integer. Thus the given continuous model has a positive solution, but the specifications are incompatible with a whole number of complete turns. This is a design-consistency conclusion, not an algebraic failure.