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definition of unit rate in mathematics - Network4

📅 May 19, 2026 👤 scraface

May 19, 2026

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📌 Use exponential growth: \( P(t) = P_0 \cdot 2^{t/T} \), where \( T = 3 \) hours.

📌 \( P(12) = 500 \cdot 2^{12/3} = 500 \cdot 2^4 = 500 \cdot 16 = 8000 \).

📌 Thus, 8000 bacteria are present after 12 hours.

📌 Solve the differential equation \( rac{dy}{dx} = 3y \) with initial condition \( y(0) = 2 \).

📌 Separate variables: \( rac{dy}{y} = 3\,dx \).

📌 Integrate both sides: \( \ln|y| = 3x + C \).

📌 Exponentiate: \( y = e^{3x + C} = Ce^{3x} \).

📌 Use \( y(0) = 2 \): \( 2 = Ce^0 \Rightarrow C = 2 \).

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