Suppose the population of a colony of bacteria doubles in 12...

Suppose the population of a colony of bacteria doubles in 12 hours from an initial population of 1 million. Find the growth constant $k$ if the population is modeled by the function $P(t)=P_{0} e^{k t} .$ When will the population reach 4 million? 8 million?

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Key Concepts

Doubling Time

Doubling time is the period required for a growing quantity to double in size. It is directly related to the growth constant by the formula k = ln(2) / (doubling time), and it is a crucial parameter for determining the dynamics of systems undergoing exponential growth.

Logarithmic Equations

Logarithmic equations are employed to solve for variables in exponential functions. In contexts like population growth, taking the natural logarithm of both sides of an equation allows one to isolate the time variable, facilitating the calculation of how long it takes for the population to reach a given level.

Exponential Growth

Exponential growth describes processes where the rate of increase is proportional to the current amount, leading to accelerated growth over time. This concept is common in modeling populations, finances, and various natural phenomena where changes compound continuously.

Growth Constant

The growth constant, denoted by k, represents the rate at which a quantity grows in the exponential model P(t) = P?e^(kt). It provides a measure of how fast the population increases and is central to linking the mathematical model with real-world growth characteristics.

Transcript

00:01 There is a colony of bacteria and at the beginning their population is 1 million, that is 10 to the 6 power.

00:16 And after 12 hours, their population becomes 2 million, that is 2 times 10 to the 6 power.

00:26 And we are looking for the growth constant k.

00:30 And let's use the formula here.

00:35 We know the formula for population.

00:39 So p, t is 12 hours, p12 equal, p0 is the initial population, e to do, k we don't know, and t is 12 hours.

00:51 After 12 hours, population becomes 2 million, that is 2 times 10 to 6, and initial population is 10 to 6.

01:00 Times e to the 12k.

01:04 And here, let me divide both sides by 10 to the 6.

01:10 So it's 10 to the 6.

01:13 So simplify them.

01:15 2 equals when we simplify e to the 12k.

01:19 And let me take both sides, natural logarithm.

01:24 Then we will get ln2 equal 12k power goes here.

01:31 L &e and we know that lne is 1.

01:36 So let me divide both sides by 12 here.

01:41 So calculate this fraction with your calculator and you will get the approximate value of k 0 .0578.

01:56 This is the approximate value of k.

02:00 I can simplify this.

02:01 And then after a while, it becomes 4 million.

02:08 The population becomes 4 million...

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