A micropaleontologist observes that the abundance $A(t)$ of a microfossil species follows $A(t) = 1000 e^{-0.03t} \sin(0.1\pi t)$, where $t$ is time in thousands of years. At what time $t > 0$ does the first local maximum occur?

["Understanding the Microfossil Abundance Function: Finding the First Local Maximum", "A micropaleontologist studying climate-driven changes in microfossil populations has modeled the abundance $ A(t) $ of a key species over time (in thousands of years) with the function:", "$$\nA(t) = 1000 e^{-0.03t} \sin(0.1\pi t)\n$$", "This hybrid model combines exponential decay (representing environmental decline or decaying population viability) with a sinusoidal term (modeling periodic environmental forcing, such as seasonal or orbital cycles). To understand critical shifts in this species’ abundance—especially when peaks occur—ecologists must identify local maxima of $ A(t) $ for $ t > 0 $. This article finds the first local maximum of $ A(t) $ by analyzing its derivative.", "---", "### Step 1: Compute the First Derivative $ A'(t) $", "To locate local maxima, we compute $ A'(t) $ and solve $ A'(t) = 0 $.", "Using the product rule:\nLet $ u(t) = 1000 e^{-0.03t} $ and $ v(t) = \sin(0.1\pi t) $. Then:", "$$\nA'(t) = u'(t)v(t) + u(t)v'(t)\n$$", "Compute each derivative:", "- $ u'(t) = 1000 \cdot (-0.03) e^{-0.03t} = -30 e^{-0.03t} $\n- $ v'(t) = \cos(0.1\pi t) \cdot 0.1\pi = 0.1\pi \cos(0.1\pi t) $", "Now substitute:", "$$\nA'(t) = (-30 e^{-0.03t}) \sin(0.1\pi t) + (1000 e^{-0.03t})(0.1\pi \cos(0.1\pi t))\n$$", "Factor out $ 1000 e^{-0.03t} $:", "$$\nA'(t) = 1000 e^{-0.03t} \left[ -0.03 \sin(0.1\pi t) + 0.1\pi \cdot 0.1 \cos(0.1\pi t) \cdot \frac{1000}{1000} \right]\n$$", "Wait — simplify constants carefully:", "Note $ 1000 \cdot 0.1\pi = 100\pi $, so:", "$$\nA'(t) = 1000 e^{-0.03t} \left( -0.03 \sin(0.1\pi t) + 0.1\pi \cos(0.1\pi t) \right)\n$$", "Alternatively, write:", "$$\nA'(t) = 1000 e^{-0.03t} \left( -0.03 \sin(0.1\pi t) + 0.1\pi \cos(0.1\pi t) \right)\n$$", "Since $ 1000 e^{-0.03t} > 0 $ for all $ t $, the sign of $ A'(t) $ depends only on the expression in parentheses.", "---", "### Step 2: Set $ A'(t) = 0 $", "Set the derivative to zero:", "$$\n-0.03 \sin(0.1\pi t) + 0.1\pi \cos(0.1\pi t) = 0\n$$", "Rearrange:", "$$\n0.1\pi \cos(0.1\pi t) = 0.03 \sin(0.1\pi t)\n$$", "Divide both sides by $ \cos(0.1\pi t) $ (valid where $ \cos <br/>\ne 0 $):", "$$\n0.1\pi = 0.03 \ an(0.1\pi t)\n$$", "Solve for tangent:", "$$\n\ an(0.1\pi t) = \frac{0.1\pi}{0.03} = \frac{10\pi}{3} \approx \frac{31.416}{3} \approx 10.472\n$$", "Now solve:", "$$\n0.1\pi t = \arctan\left( \frac{10\pi}{3} \right) + n\pi, \quad n \in \mathbb{Z}\n$$", "Compute $ \arctan\left( \frac{10\pi}{3} \right) \approx \arctan(10.472) $. Since $ \ an(\pi/2) \ o \infty $, and $ 10.472 \gg 1 $, $ \arctan(10.472) \approx 1.471 $ radians (using calculator or known approximation).", "So:", "$$\n0.1\pi t \approx 1.471 \quad \Rightarrow \quad t \approx \frac{1.471}{0.1\pi} \approx \frac{1.471}{0.3142} \approx 4.69\n$$", "But we must consider the first positive solution. Since $ \arctan $ returns values in $ (-\pi/2, \pi/2) $, the smallest positive solution comes from $ n = 0 $:", "$$\n0.1\pi t = \arctan\left( \frac{10\pi}{3} \right)\n\quad \Rightarrow \quad\nt = \frac{1}{0.1\pi} \arctan\left( \frac{10\pi}{3} \right)\n$$", "Let’s compute more precisely:", "- $ \frac{10\pi}{3} \approx 10.47197551 $\n- $ \arctan(10.47197551) \approx 1.47118 $ radians\n- $ 0.1\pi \approx 0.314159 $\n- So $ t \approx \frac{1.47118}{0.314159} \approx 4.687 $", "Now verify it’s a maximum.", "---", "### Step 3: Confirm It Is a Local Maximum", "We analyze the sign of $ A'(t) $ around $ t \approx 4.687 $. For $ t < 4.687 $, $ 0.1\pi t < 1.471 $, so $ \ an(0.1\pi t) < \frac{10\pi}{3} $, hence:", "$$\n0.1\pi t < \arctan\left( \frac{10\pi}{3} \right) \Rightarrow \ an(0.1\pi t) < \frac{10\pi}{3}\n\Rightarrow 0.1\pi \cos(0.1\pi t) - 0.03 \sin(0.1\pi t) > 0\n\Rightarrow A'(t) > 0\n$$", "For $ t > 4.687 $, $ \ an(0.1\pi t) > \frac{10\pi}{3} $, so the expression becomes negative, hence $ A'(t) < 0 $.", "Thus, $ A(t) $ increases before $ t \approx 4.687 $ and decreases afterward → local maximum.", "---", "### Step 4: Final Answer", "The first local maximum of $ A(t) $ occurs at:", "$$\nt = \frac{1}{0.1\pi} \arctan\left( \frac{10\pi}{3} \right) = \frac{10}{\pi} \arctan\left( \frac{10\pi}{3} \right)\n$$", "Numerically, this is approximately $ t \approx 4.687 $, but the exact form is preferred.", "However, since the problem asks “At what time $ t > 0 $”, and expects a precise answer suitable for scientific interpretation, we box the exact expression:", "$$\n\boxed{t = \frac{10}{\pi} \arctan\left( \frac{10\pi}{3} \right)}\n$$", "This is the first time after $ t = 0 $ at which the microfossil abundance reaches a local peak, reflecting a biologically significant oscillation modulated by long-term environmental change.", "---", "### Why This Matters", "This peak corresponds to a period of elevated abundance driven by a sinusoidal environmental factor—perhaps climate cycles—while being dampened over time by exponential decay. Identifying such maxima helps micropaleontologists correlate microfossil records with paleoclimatic proxies, improving our understanding of ecosystem resilience and response to past global changes.", "---", "Keywords: micropaleontologist, $ A(t) = 1000 e^{-0.03t} \sin(0.1\pi t) $, local maximum, exponential decay, sinusoidal abundance, paleoclimate modeling, first peak time."]