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Chapter 24: Problem 33

Suppose that an organism replicates itself each second. If you start with a single specimen, what will the final population be after 10 seconds?

Short Answer

Expert verified
The final population after 10 seconds is 1024.

Step by step solution

01

Identify Initial Population

Start with the initial number of specimens. In this case, we start with 1 specimen.
02

Understand Replication Process

The organism replicates itself every second. This means that the population doubles every second.
03

Calculate Population After Each Second

For each second, calculate the new population by doubling the previous population. This can be formulated as: \[ P_n = 2^n \] where \(P_n\) is the population at second \(n\).
04

Apply the Formula

To find the population after 10 seconds, substitute 10 for \(n\). This gives: \[ P_{10} = 2^{10} \]
05

Calculate the Final Population

Evaluate the expression \(2^{10}\). \[ 2^{10} = 1024 \]

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Key Concepts

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

exponential growth
Population growth can occur in many ways, but one of the most fascinating is exponential growth. Exponential growth happens when the rate of population increase is proportional to the current population. This means that the larger the population gets, the faster it grows. In our case, the organism doubles its population every second.
Imagine starting with just one organism. After 1 second, you have 2 organisms. After 2 seconds, you have 4 organisms. This doubling continues until it seems like the population is growing incredibly fast!
replication process
The replication process is how a population grows over time. In our example, the organism replicates itself every second. This means that each existing organism creates one more organism per second.
  • At the start (0 seconds), you have just 1 organism.
  • After 1 second, each organism doubles, so you have 2 organisms in total.
  • After 2 seconds, you have 4 organisms.

This doubling process continues, with the population rapidly increasing. This is why exponential growth can lead to very large populations in a short amount of time.
mathematical formula
To understand the growth precisely, we use a mathematical formula. The formula for exponential growth when an organism doubles every second is \( P_n = 2^n \). Here, \( P_n \) is the population at second \( n \).
Let's calculate the population after 10 seconds with this formula. We substitute 10 for \( n \), giving us \[ P_{10} = 2^{10} \].
When we simplify \[ 2^{10} = 1024 \], we find that the final population after 10 seconds is 1024 organisms.
This formula is powerful because it quickly tells us how large the population will be after a specific number of seconds, making it easy to predict future growth.

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Most popular questions from this chapter

Scientists look for water to indicate places where life might exits because a. water is a common molecule in interstellar space. b. life on Earth depends on it. c. no other molecules are solvents. d. the spectrum of water is very complex.

A thermophile is an organism that lives in extremely __________ environments. a. salty b. hot c. cold d. dry

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?

The Urey-Miller experiment produced _________ in a laboratory jar. a. life b. RNA and DNA c. amino acids d. proteins

The study of life and the study of astronomy are connected because (select all that apply) a. life may be commonplace in the universe. b. studying other planets may help explain why there is life on Earth. c. explorations of extreme environments on Earth suggest where to look for life elsewhere. d. life is a structure that evolved through physical processes, and life on Earth may not be unique. e. life elsewhere is most likely to be found by astronomers.
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