Question
A bacterial colony grows at a rate proportional to its current population. If the population doubles in 3 hours, by what factor does it grow in 9 hours?
Solution — Step by Step
“Rate proportional to current population” translates to:
where is the population and is the growth constant.
Integrate: , or where is the initial population.
At , :
, so .
.
The population grows 8× in 9 hours.
Why This Works
Exponential growth is the unique solution to “rate proportional to amount.” Every doubling time multiplies the population by 2 — three doublings in 9 hours give .
This logic applies to compound interest, radioactive decay (with negative ), Newton’s law of cooling, and population dynamics.
Speed shortcut: For exponential growth, “factor in time ” , where is the doubling time. Skip the differential equation entirely for problems like this.
Alternative Method — Direct Ratio
Since growth is exponential, the factor in time is . The factor in hours is .
This avoids solving for explicitly — just exploit the exponential structure.
Common Mistake
Students often write linear growth: “If it doubles in 3 hours, it triples in 9 hours” — wrong. Linear growth is constant addition; exponential growth is constant multiplication.
Another trap: confusing doubling time with half-life formulas. For decay, where is half-life. Same structure, just with instead of .
CBSE Class 12 boards consistently include a “growth and decay” question for 4-6 marks. JEE Main rephrases this as Newton’s law of cooling or radioactive decay. The math is identical — it is just word problem disguise.
For a real-world twist: bacterial growth doesn’t continue forever (resources run out). The logistic equation models this saturation, but JEE/NEET stop at the simple exponential model.