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Easy: Predicting growth
When a quantity changes at a rate proportional to its size, dy/dt = ky. Separating variables and integrating gives y = y₀e^(kt), where y₀ is the starting amount. k > 0 means growth and k < 0 decay. If the amount doubles in time T, then e^(kT) = 2, so k = ln 2 ÷ T.
Use y = y₀e^(kt). Round to the nearest hundredth.
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It says the rate of growth is proportional to the current amount, so the more there is, the faster it grows. The function with that property is y = y₀e^(kt).
Set y = 2y₀ at the doubling time T: 2 = e^(kT), so k = ln 2 / T. For a half-life, k = −ln 2 / T.
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