Repeated scaling makes a curve

1 · Learn

A simulation starts with 10 units of a quantity and doubles it every hour. Its continuous model is Q = 10 × 2^t, where t is hours since the start. At t = 0, 1, 2 and 3, Q is 10, 20, 40 and 80 units. Equal time steps multiply by 2; the added amounts are 10, 20 and 40, so this is not a straight-line increase.

The supplied curve represents the stated continuous model between those whole-hour times too. A sequence that only records completed steps would instead use separate points. Do not assume that every real process follows either model just because a few values agree.

To estimate when Q reaches 30, read horizontally from 30 to the curve, then down to time. It is between 1 and 2 hours, about 1.6 hours. Use a calculator to check nearby inputs if needed: at 1.5 hours Q is about 28.3, whereas at 1.6 hours it is about 30.3. No logarithm method is needed for this graphical estimate.

Repeated shrinking is exponential too. For Q = 16 × (1/2)^t, the whole-hour values are 16, 8, 4 and 2. The curve falls but stays positive for every finite non-negative time. It does not decrease by a constant amount or cross zero. Keep a model's assumptions and time range explicit.

2 · Worked example

In Q = 10 × 2^t, the graph reaches 40 units at t = 2 hours. At t = 2.5, the curve gives about 57 units; 10 × 2^2.5 is about 56.6. Halfway in time is not halfway between 40 and 80: that straight-line estimate would be 60.

3 · Your turn

Read the graph for Q at t = 1 and for the time when Q is 60. Explain why joining (0,10) straight to (3,80) is wrong. In the decay model 16 × (1/2)^t, find Q at t = 3.

4 · Apply your learning

Plot the decay model's values at t = 0, 1, 2, 3 on squared paper and sketch a smooth falling curve. Mark when it reaches 6 units approximately. State why neither model alone predicts what a real experiment must do.