Proportional rates, half-life and model limits
| English | 中文 | Pinyin |
|---|---|---|
| proportional rate | 比例变化率 | bǐ lì biàn huà lǜ |
A decaying amount loses more per hour when more remains. Why does one fixed half-life not mean a fixed subtraction?
- A decaying amount loses more per hour when more remains. Why does one fixed half-life not mean a fixed subtraction?
- This lesson studies proportional rate 比例变化率: A rate of change equal to a constant multiplied by the current amount.
Choose the mathematical structure
- For y=Ae^(kt), dy/dt=ke^(kt)A=ky. A>0 is the initial amount; k is a constant proportional rate in inverse time units. Positive k gives growth, negative k decay, zero k a constant. Over elapsed time Δt the multiplier is e^(kΔt). Doubling time is ln2/k for k>0; half-life is ln2/|k| for k<0.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines proportional rate?
A rate of change equal to a constant multiplied by the current amount.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
Use an illustrative decay Q=80e^(−λt), t in hours and λ=ln2/4. Then Q₄=40, Q₈=20 and Q₁₂=10: each four-hour step halves the amount. At t=4 the rate is Q′=−40λ=−10ln2 amount units per hour. Solving Q≤10 gives t≥12; the strict condition Q<10 instead needs t>12. For an illustrative mathematical account P=100e^(0.1t), t in years, P₁≈110.517 and the one-year effective growth is about 10.517%, not 10%. If the given effective yearly growth is 10%, use k=ln1.1 instead. Continuous compounding and a stated annual multiplier must not be confused.
Proportional rates, half-life and model limits
For y=Ae^(kt), dy/dt=ke^(kt)A=ky
State the valid input, plotted variable or time unit before using the logarithmic/exponential relationship.
Find Q at t=4 hours.
80e^(−ln2)=40.
Test a tempting shortcut
- The exponent must be dimensionless: changing hours to minutes changes k by a factor 1/60. A half-life is not the time to reach zero; positive A times an exponential stays positive for finite time. Absolute losses vary with the amount although the proportional rate is fixed. A first whole-period threshold needs adjacent integer checks rather than only a continuous crossing time.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A constant exponential proportional rate means the same absolute amount is lost in each equal time interval. This claim is false. Explain which definition or assumption it violates.
Find Q at t=8 hours.
80e^(−2ln2)=20.
A constant exponential proportional rate means the same absolute amount is lost in each equal time interval.
The exponent must be dimensionless: changing hours to minutes changes k by a factor 1/60. A half-life is not the time to reach zero; positive A times an exponential stays positive for finite time. Absolute losses vary with the amount although the proportional rate is fixed. A first whole-period threshold needs adjacent integer checks rather than only a continuous crossing time.
Interpret a new situation
- State time/amount units, initial conditions and the constant-rate assumption. Compare observations with predictions before extrapolating. Limited resources, changing environmental conditions, fees or withdrawals can invalidate a simple growth model; changing decay conditions can require a varying-rate or multi-stage model. These are illustrative mathematical models, not product terms or treatment instructions.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the first time in hours when Q≤10.
ln(80/10)/(ln2/4)=12; monotone decay gives Q≤10 from that time onward.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · F. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A rate of change equal to a constant multiplied by the current amount. Choose the relationship, show the method, check its assumptions and interpret the result.