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Home » You Won’t Believe What Zurg Can Do—This Creative Monster is Spinning Stars! - 500apps

You Won’t Believe What Zurg Can Do—This Creative Monster is Spinning Stars! - 500apps

📅 March 15, 2026 👤 scraface

Mar 15, 2026

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📌 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.

📌 Solution: Set $ 20 + 80e^{-0.1t} = 40 $. Subtract 20: $ 80e^{-0.1t} = 20 $. Divide by 80: $ e^{-0.1t} = 0.25 $. Take natural logarithm: $ -0.1t = \ln(0.25) $. Solve: $ t = -\frac{\ln(0.25)}{0.1} = \frac{\ln(4)}{0.1} \approx \frac{1.386}{0.1} = 13.86 $. Round to two decimal places: $ \boxed{13.86} $ minutes.The Analytical Engine, designed by Charles Babbage, is considered a precursor to modern computers. If Babbage’s design allows a program to execute 120 distinct operations and each operation requires 3 memory accesses, how many total memory accesses are needed to complete the program once?

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