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📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 2^{n-1} > \frac{500}{3} \approx 166.67

📌 Now compute powers of 2:

📌 $ 2^7 = 128 $, $ 2^8 = 256 $

📌 So $ 2^{n-1} > 166.67 $ implies $ n-1 \geq 8 $, thus $ n \geq 9 $.

📌 On day 9, the count is $ 3 \cdot 2^8 = 3 \cdot 256 = 768 > 500 $.

📌 On day 8: $ 3 \cdot 2^7 = 384 < 500 $, so day 9 is the first.

📌 Question: A self-healing polymer film repairs itself over time, with the repair rate modeled by $ r(t) = \sqrt{(t - 8)^2} $ millimeters per hour. For how many integer values of $ t $ is the repair rate exactly 4 mm/hour?

📌 Solution: We are given $ r(t) = \sqrt{(t - 8)^2} $.

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