Model doubling, recover initial counts and convert bases
Interpret continuous growth and compare scenarios
Calibrate and validate two-point exponential models
Build half-life decay models with unit conversions
Infer half-life, remaining fractions and original mass
Estimate isotope ages and state assumptions
Calibrate and interpret Newton cooling models
Model doubling, recover initial counts and convert bases · Worked example
A simulated culture begins with 320 organisms and doubles every 60 minutes. With \(t\) measured in hours, derive its base-2 and base-e models, calculate the count at 1.75 hours, solve the time to 2240 organisms, and plot the model through that target. Explain the difference between a continuous estimated count and an actual integer count. Show written work; selecting a summary alone is not a complete response. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
One doubling takes 60 minutes, or \(a=\dfrac{60}{60}=1\) hours. After \(t\) hours the number of doubling periods is \(\dfrac{t}{a}\). \(N_0=320\). \(N(t)=N_0\,2^{\dfrac{t}{a}}=N_0 e^{(\ln 2)\dfrac{t}{a}}\). Thus \(r=\dfrac{\ln 2}{1}\)\(\approx 0.693147\) per hour. The base-e and base-2 graphs coincide exactly when the unrounded rate is used. \(N(1.75)=N_0\,2^{\dfrac{1.75}{1}}\approx 1076.35\) organisms; round only the final estimated count to 1076. \(N_0e^{rt}=2240\Rightarrow rt=\ln\left(\dfrac{2240}{N_0}\right)\Rightarrow t=\dfrac{\ln(\dfrac{2240}{N_0})}{r}\approx 2.8074\) hours from introduction. Physical domain: \(t\ge0\). The continuous curve estimates an integer population and assumes an unchanged doubling time and no resource limit. It does not assert a fractional organism or indefinite real-world growth. Reference graph. Open circles: observations or key values. Solid: both equivalent models. Written-work review:
Convert the 60-minute doubling time to hours. Derive both the base-two and base-e models, with the natural-log conversion giving a positive rate per hour.
Use 320 organisms as the initial count, not as a fixed additive increase per doubling period.
Evaluate the model at total time 1.75 hours and solve logarithmically for the time to 2240 organisms. Label the target time as hours since introduction, not additional hours from the present.
Draw an increasing exponential curve with labeled hour and organism axes, the initial intercept, forecast point and target point. Include the whole requested time interval and explain why the two exact model forms have the same graph.
State the nonnegative-time physical domain and unchanged-rate and resource assumptions. Distinguish a continuous estimated population from a literal fractional organism; round to a whole count only at the reporting stage.
Numerical grading: accept equivalent exact expressions and final values within one unit in the last decimal displayed in the worked solution; for scientific notation apply that tolerance to the mantissa, and allow nearest-whole counts when requested. Preserve intermediate precision, stated units and interval endpoints. Assess each derivation and explanation separately; a selected summary alone is not a complete response.