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Home » \( 0 = 20 - 9.8t \) - High Altitude Science

\( 0 = 20 - 9.8t \) - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 5Question: Find the time $ t $ at which the growth rates of two marine species, modeled by $ 3^{t} $ and $ 9^{t-1} $, are equal.

📌 Solution: Set $ 3^{t} = 9^{t-1} $. Since $ 9 = 3^{2} $, rewrite the equation as $ 3^{t} = (3^{2})^{t-1} $. Simplify the right-hand side: $ 3^{t} = 3^{2(t-1)} $. Equate the exponents: $ t = 2(t - 1) $. Solve: $ t = 2t - 2 $, so $ t = 2 $. Verify by substituting back: $ 3^{2} = 9^{1} $ → $ 9 = 9 $. Thus, the time is $ \boxed{2} $.

📌 Question: A community biotech researcher mixes two solutions: one with 15% vinegar and another with 5% vinegar. How many liters of each are needed to make 20 liters of 10% vinegar?

📌 Solution: Let $ x $ be liters of 15% solution and $ y $ be liters of 5% solution. The system is:

📌 x + y = 20 \\

📌 0.15x + 0.05y = 2 \quad (\text{since } 10\% \text{ of } 20 = 2)

📌 From the first equation, $ y = 20 - x $. Substitute into the second: $ 0.15x + 0.05(20 - x) = 2 $. Simplify: $ 0.15x + 1 - 0.05x = 2 $ → $ 0.10x = 1 $ → $ x = 10 $. Then $ y = 10 $. Thus, $ \boxed{10} $ liters of each solution are needed.

📌 Question: A historian of science studies the cooling of a 17th-century thermometer, modeled by $ T(t) = 20 + 80e^{-0.1t} $. Find the time $ t $ when the temperature reaches 40°C.

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