For the following exercises, use this scenario: A biologist...

Name: Problem 411, Chapter 4: Precalculus
Uploaded: 2020-10-02T14:24:58-08:00
Duration: 4 min 25 s
Channel: Arielle Vanarsdall
Description: For the following exercises, use this scenario: A biologist recorded a count of 360 bacteria present in a culture after 5 minutes and 1000 bacteria present after 20 minutes. Rounding to six significant digits, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?

For the following exercises, use this scenario: A biologist recorded a count of 360 bacteria present in a culture after 5 minutes and 1000 bacteria present after 20 minutes. Rounding to six significant digits, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?

For the following exercises, use this scenario: A biologist recorded a count of 360 bacteria present in a culture after 5 minutes and 1000 bacteria present after 20 minutes.

Rounding to six significant digits, write an exponential equation representing this situation. To the nearest minute, how long did it take the population to double?

Key Concepts

Exponential Growth

Exponential growth refers to the process where a quantity increases at a rate proportional to its current value. In biology, this is commonly modeled by an equation of the form P(t) = P? · e^(kt), where P(t) is the population at time t, P? is the initial population, and k is the growth constant. This model is especially relevant for populations like bacteria under ideal conditions where resources are abundant.

Growth Constant Estimation

Determining the growth constant (k) involves using data points from the exponential growth model. Given two measurements at different times, one can solve for k by taking the natural logarithm of the ratio of the populations and dividing by the time difference between the observations. This step is crucial for tailoring the exponential equation to the observed behavior of the population.

Doubling Time

Doubling time is the period required for a quantity undergoing exponential growth to double in size. Mathematically, it is obtained using the formula T?double? = ln(2) / k, where k is the growth constant. This concept not only helps in understanding the dynamics of population growth, but it also simplifies comparisons between different exponential growth scenarios.

Rounding and Significant Digits

When expressing numerical values in an exponential equation, it is important to round the parameters to the required number of significant digits to maintain the desired precision. This practice ensures consistency and accuracy in calculations and communication of results, especially when specific rounding instructions, such as six significant digits, are provided in a problem.

Transcript

00:01 All right, for this equation, we're going to try to figure out if we can set up an equation to show how quickly the bacteria grows.

00:08 And then we also want to know how quickly it takes to double.

00:13 So given what we know, which is not the initial value, we're going to use our data points that at five minutes, we have 360 bacteria, and at 20 minutes, we have 1 ,000.

00:22 We'll set up some equations to help us find the rate and the initial value.

00:25 To begin, 360 equals initial value a, ulor's constant of 5 times the rate.

00:35 We're going to solve for a here in dividing both sides, and we get a equals 360 e to the negative 5r.

00:48 We have negative 5 so that we can put our ular's constant with an exponent in the numerator.

00:54 Now hang on to this over here, and then go make our second essay.

01:00 Over here we have 1 ,000 equals are the value we just found for a, 625r times e2 .r.

01:14 Knowing our rules here, we can combine these terms, 1 ,000, plus 310e to the 15r, side both sides of 360, and we get approximately 2 .778, and they ask us for six significant digits.

01:34 We can go there.

01:38 Using a natural log on both sides, we can solve.

01:48 We know that the natural log of e is just one.

01:51 And the natural log of this value here is 1 .25.

01:59 And we'll divide both sides 16.

02:03 And our approximate value for r is 0 .068110...

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