Simulation · Maths · Class 12

How long to double?

From the lesson Growth, decay and cooling in Differential Equations. Change the values and watch what happens.

The idea behind it

NCERT §9.4.1 (Example 9, applications)

If a quantity grows at a rate proportional to its size, dP/dt = kP with k > 0. Separating gives P = P₀e^(kt), where P₀ is the value at t = 0.
The time for P to double is found from 2 = e^(kT): T = (log 2)/k, the same from any starting value.
Worked example: money grows continuously at 10% a year, so k = 0.1. It doubles in 10 log 2 ≈ 6.93 years, and Rs 1000 becomes 1000e ≈ Rs 2718 after 10 years. At 7% a year it doubles in about 9.9 years.
Decay is the same equation with a negative constant: dN/dt = −λN gives N = N₀e^(−λt) and a half-life of (log 2)/λ.
Worked example: a culture that triples in 5 hours has e^(5k) = 3. After 10 hours it is e^(10k) = 3² = 9 times its starting size.
Worked example (cooling): a drink at 85 °C in a 25 °C room follows dT/dt = −k(T − 25), so T − 25 = 60e^(−kt). If it reaches 55 °C in 10 minutes, the gap 60 halves in 10 minutes, so after 20 minutes the gap is 15 and the drink is at 40 °C.

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