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MathCalculus

Exponential Function Formula: Bacteria Doubling Every 12 Hours

Use Q = Q₀ e^(kt) to find the growth constant for a colony starting at 1,000 and doubling every 12 hours, then find when it reaches 200,000.

Question

OpenStax Calculus Volume 1, Section 1.5, Exercise 307 [T] (standardized wording).

A laboratory bacterial population starts at 1,000 and doubles every 12 hours. (a) In the exponential model Q=Q0ektQ=Q_0 e^{kt}, calculate the bacterial growth-rate constant k and report it to four decimal places. (b) Estimate the elapsed time until the population reaches 200,000 bacteria.

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Evidence boundary

The prompt is a standardized restatement of OpenStax Calculus Volume 1 Section 1.5 Exercise 307, including both parts and the four-decimal instruction. The population follows the stated ideal exponential model with t in hours. The target time is calculated with unrounded k; the alternative using k = 0.0578 is identified explicitly. These are model calculations, not observations of a real culture.