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america's got talent free - RTA

📅 May 10, 2026 👤 scraface

May 10, 2026

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📌 a^3 + b^3 = 12^3 - 3(27)(12) = 1728 - 972 = 756

📌 Thus, the sum of the cubes of lesson plans and activities contributions is 756.

📌 \boxed{756}

📌 Question: In a biology lab, a culture grows such that the number of cells triples every hour. If the population after 4 hours is $ 3^6 $ times the initial count, verify the growth model and compute the number of cells after 4 hours if the starting amount is 50.

📌 Solution: The model states that each hour the count is multiplied by 3. After 1 hour: $ 3^1 $, after 2 hours: $ 3^2 $, so after 4 hours: $ 3^4 $. But the problem states the population is $ 3^6 $ times initial, which contradicts unless growth rate differs. However, assuming the model is correct, after 4 hours:

📌 \text{Population} = 50 \times 3^4 = 50 \times 81 = 4050

📌 But the problem says $ 3^6 = 729 $ times original — so unless additional doubling occurs, the model may be misstated. But based on the stated exponential model $ P(t) = P_0 \cdot 3^t $, after 4 hours:

📌 P(4) = 50 \cdot 3^4 = 50 \cdot 81 = 4050

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