Question: A museum curator uses a digital model where the position $ s(t) $ of a rotating exhibit arm is given by $ s(t) = t - \frac{t^5}{5} $. Define $ c_n $ by $ c_1 = \frac{1}{2} $ and $ c_{n+1} = s(c_n) $. Compute $ c_3 $.

["Title: Computing the Third Iterate of a Curator’s Digital Rotating Exhibit Model\nMeta Description: Explore how a museum curator models a rotating exhibit arm using a digital function. Learn to compute $ c_3 $ when $ s(t) = t - \frac{t^5}{5} $ and $ c_1 = \frac{1}{2} $.\nKeywords: digital modeling, rotating exhibit arm, museum curator, mathematical iteration, $ c_3 $ computation, $ s(t) = t - \frac{t^5}{5} $", "---", "A fascinating intersection of art, technology, and mathematics lies in how modern museums use digital simulations to design dynamic exhibits—such as rotating arms that captivate visitors. One such example involves a digital model where the motion of an exhibit arm is defined by the function:\n$$ s(t) = t - \frac{t^5}{5} $$\nFor precise control, a curator defines a recursive sequence to simulate progressive rotations:\n- $ c_1 = \frac{1}{2} $\n- $ c_{n+1} = s(c_n) $", "In this article, we compute $ c_3 $ to illustrate how iterative application of this smooth, non-linear function models the arm’s evolving position. Understanding $ c_3 $ reveals how small initial positions transform under repeated digital transformations, showcasing both mathematical elegance and real-world applicability in museum design.", "---", "### Step 1: Compute $ c_2 $\nGiven $ c_1 = \frac{1}{2} $, apply the function $ s $:\n$$\nc_2 = s(c_1) = s\left(\frac{1}{2}\right) = \frac{1}{2} - \frac{1}{5} \left(\frac{1}{2}\right)^5\n$$\nCalculate $ \left(\frac{1}{2}\right)^5 = \frac{1}{32} $:\n$$\nc_2 = \frac{1}{2} - \frac{1}{5} \cdot \frac{1}{32} = \frac{1}{2} - \frac{1}{160} = \frac{80}{160} - \frac{1}{160} = \frac{79}{160}\n$$", "---", "### Step 2: Compute $ c_3 $\nNow use $ c_2 = \frac{79}{160} $ and apply $ s $ again:\n$$\nc_3 = s(c_2) = c_2 - \frac{c_2^5}{5} = \frac{79}{160} - \frac{1}{5} \left(\frac{79}{160}\right)^5\n$$", "First, compute $ \left(\frac{79}{160}\right)^5 $. Since this is a small fractional power of a number less than 1, the result is very small. For accuracy, we calculate step-by-step:", "- $ \frac{79}{160} = 0.49375 $\n- $ (0.49375)^5 \approx 0.0289 $ (using calculator or approximation)\n More precisely:\n $$\n 0.49375^2 \approx 0.2437,\quad 0.49375^4 \approx (0.2437)^2 \approx 0.0594,\quad 0.49375^5 = 0.0594 \ imes 0.49375 \approx 0.02935\n $$\n Thus,\n $$\n \frac{1}{5} \left(\frac{79}{160}\right)^5 \approx \frac{0.02935}{5} = 0.00587\n $$", "Now compute:\n$$\nc_3 = 0.49375 - 0.00587 = 0.48788\n$$", "---", "### Final Result:\n$$\nc_3 = \frac{79}{160} - \frac{1}{5} \left(\frac{79}{160}\right)^5 \approx 0.48788\n$$\nFor exact symbolic form, retain fractional exponents:\n$$\nc_3 = \frac{79}{160} - \frac{1}{5} \left(\frac{79}{160}\right)^5\n$$\nBut numerically, $ c_3 \approx 0.4879 $, demonstrating how nonlinear dynamics gently reduce the arm’s position over iterations—ideal for smooth, controlled motion in museum displays.", "---", "This calculation exemplifies how curators and digital artists leverage mathematical models to bring exhibits to life, blending physics, computation, and aesthetics into seamless visitor experiences.", "---", "Key Takeaways:\n- $ c_1 = \frac{1}{2} $\n- $ c_2 = \frac{79}{160} \approx 0.49375 $\n- $ c_3 = s(c_2) \approx 0.48788 $\n- The function $ s(t) = t - \frac{t^5}{5} $ provides smooth, dampened motion ideal for digital rotating displays.", "For museum designers and enthusiasts, such models bridge clinical mathematics with mesmerizing visual storytelling.", "Related Topics:\n- How mathematical functions shape digital museum exhibits\n- Iterative simulation in kinetic art installations\n- Numerical approximation techniques in curatorial technology", "---\nThe digital rotation of museum exhibits starts with elegant functions—and a simple recursive formula."]