4.6 Modeling with Exponential Functions

A simulated colony was introduced $3.75$ hours ago and now has an estimated $1625$ organisms. Its doubling time is $45$ minutes. Take $t=0$ at introduction. Recover the initial count without early rounding, write equivalent base $-2$ and base $-e$ models with time in hours, predict the count $1$ hours from now, find when the count reaches $4062.5$, and sketch the model from introduction through that target. 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 $45$ minutes, or $a=\dfrac{45}{60}=0.75$ hours. After $t$ hours the number of doubling periods is $\dfrac{t}{a}$.
$N_0=\dfrac{1625}{2^{\dfrac{3.75}{0.75}}}\approx 50.78125$ organisms. Retain this exact expression in later work.
$N(t)=N_0\,2^{\dfrac{t}{a}}=N_0 e^{(\ln 2)\dfrac{t}{a}}$. Thus $r=\dfrac{\ln 2}{0.75}$$\approx 0.924196$ per hour. The base $-e$ and base $-2$ graphs coincide exactly when the unrounded rate is used.
$N(4.75)=N_0\,2^{\dfrac{4.75}{0.75}}\approx 4094.74$ organisms; round only the final estimated count to $4095$.
$N_0e^{rt}=4062.5\Rightarrow rt=\ln\left(\dfrac{4062.5}{N_0}\right)\Rightarrow t=\dfrac{\ln(\dfrac{4062.5}{N_0})}{r}\approx 4.7414$ 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.

• Convert the $45-$ 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.
• Recover the introduction count by dividing the count at hour $3.75$ by its elapsed growth factor. Keep that recovered value unrounded; do not reset time zero to the present.
• Evaluate the model at total time $4.75$ hours and solve logarithmically for the time to $4062.5$ 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.