C: $2\pi\sqrt{1 + \frac{1}{16\pi^2}}$

February 22, 2026

$C: $2\pi\sqrt{1 + \frac{1}{16\pi^2}}$$

Related Articles

Question: A geologist studying cave formations observes that a stalactite grows in a spiral path modeled by the parametric equations $x(t) = \cos t$, $y(t) = \sin t$, $z(t) = \frac{t}{4\pi}$, where $t \geq 0$. Find the arc length of the stalactite's growth from $t = 0$ to $t = 4\pi$ years.
A: $\int_0^{4\pi} \sqrt{\sin^2 t + \cos^2 t + \left(\frac{1}{4\pi}\right)^2} \, dt = 4\pi \sqrt{1 + \frac{1}{16\pi^2}}$
B: $4\pi$ meters
D: $4\pi \sqrt{1 + \frac{1}{16\pi^2}}$
Answer: A
Question: A micropaleontologist analyzing oxygen isotope ratios in foraminifera uses the function $I(t) = 3\cos\left(\frac{\pi t}{6}\right) + 4\sin\left(\frac{\pi t}{6}\right)$, where $t$ is time in thousands of years. Find the maximum value of $I(t)$ over all $t \in \mathbb{R}$.

Trending Articles