C: $2\pi\sqrt{1 + \frac{1}{16\pi^2}}$
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- 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 $t \in \mathbb{R}$.
