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Chapter 24: Problem 34

Suppose that an organism replicates itself each second. After how many seconds will the population increase by a factor of \(1,024 ?\)

Short Answer

Expert verified
10 seconds

Step by step solution

01

Understand the Problem

The organism replicates itself every second, doubling its population each time. Given that the population increases by a factor of 1,024, we need to find how many seconds this process takes.
02

Express Population Growth

If the initial population is represented as 1, after 1 second it becomes 2, after 2 seconds it becomes 4, and so on. Mathematically, this can be expressed by the formula: \[ P = 2^n \]where \( P \) is the population after \( n \) seconds.
03

Set Up the Equation

According to the problem, the population increases by a factor of 1,024. Thus, we set up the equation: \[ 1,024 = 2^n \]
04

Solve for \( n \)

To find \( n \), we need to determine how many times 2 must be multiplied by itself to equal 1,024. This can be solved using properties of exponents or logarithms.First, recognize that 1,024 is a power of 2:\[ 2^{10} = 1,024 \]So, \( n = 10 \).
05

Conclusion

It takes 10 seconds for the population to increase by a factor of 1,024.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Replication Rate
Replication rate is the frequency at which an organism reproduces itself. In this example, the organism replicates every second. This means every second, the population doubles. So, if you start with one organism:
- After 1 second, you have 2 organisms.
- After 2 seconds, you have 4 organisms.
- After 3 seconds, you have 8 organisms.
This repeating doubling process defines the replication rate as 1 replication per second. Understanding the replication rate is crucial to predict how quickly a population can grow.
Population Doubling Time
Population doubling time tells us how long it takes for a population to double in size. In our problem, the doubling happens every second, so the doubling time is 1 second. If you know the doubling time and the initial population, you can calculate future population sizes. For example:
- Start with 1 organism and a doubling time of 1 second.
- After 1 second, the population is 2 (doubled from 1).
- After 2 seconds, the population is 4 (doubled from 2).
The exponential growth continues until a significant population size is reached. This concept is especially useful in fields like biology and environmental science to model growth patterns.
Exponential Functions
Exponential functions are mathematical expressions used to model growth processes. In this case, we use the exponential function to describe how the population grows over time. The general form is:
\[ P = 2^n \]
Where:
  • P is the population at time n (number of seconds).
  • 2 is the base representing the doubling effect.
  • n is the number of time intervals (seconds).
To solve the problem of finding when the population reaches 1,024, you set up the equation:
\[ 1,024 = 2^n \]
Recognize that 1,024 is equal to \( 2^{10} \). Therefore, it takes 10 seconds for the population to grow to the desired size. This illustrates how exponential functions provide a powerful way to model and predict rapid growth.

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Most popular questions from this chapter

Life first appeared on Earth a. billions of years ago. b. millions of years ago. c. hundreds of thousands of years ago. d. thousands of years ago.

In the phrase "theory of evolution," the word theory means that evolution a. is an idea that can't be tested scientifically. b. is an educated guess to explain natural phenomena. c. probably doesn't happen anymore. d. is a tested and corroborated scientific explanation of natural phenomena.

The doubling time for Escherichia coli is 20 minutes, and you start getting sick when just 10 bacteria enter your system. How many bacteria are in your body after 12 hours?

The search for life elsewhere in the Solar System is carried out primarily by a. astronauts. b. robotic spacecraft. c. astronomers using optical telescopes. d. astronomers using radio telescopes.

As a general rule, you can find the doubling time for exponential growth by dividing 70 by the rate of increase. So, if the population increases by 7 percent per year, the doubling time is 10 years \((70 / 7=10) .\) Suppose Earth's human population continues to grow by 1 percent annually. What is the doubling time? How much time will pass before there are 4 times as many humans on Earth?
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