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Chapter 24: Problem 35

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?

Short Answer

Expert verified
The doubling time is 70 years. It takes 140 years for the population to become four times as many.

Step by step solution

01

Understand the Doubling Time Formula

The doubling time for exponential growth can be found by dividing 70 by the rate of increase. The formula is: \[ \text{Doubling Time} = \frac{70}{\text{Rate of Increase (\text{%})}} \]
02

Calculate Doubling Time

Given the rate of increase is 1 percent per year, plug this value into the formula to find the doubling time:\[ \text{Doubling Time} = \frac{70}{1} \] This simplifies to 70. So, the doubling time is 70 years.
03

Determine Time to Become Four Times

If the population doubles every 70 years, we need to determine how long it will take to become four times the original population. To quadruple, the population must double twice ((2^2 = 4)).Thus, the time taken to quadruple the population is:\[ 70 \text{ years} + 70 \text{ years} = 140 \text{ years} \]

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Key Concepts

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

Doubling Time Formula
The doubling time formula is a crucial concept in understanding exponential growth. It's fairly straightforward: To find out how long it takes for something (like a population) to double, you divide 70 by the growth rate (expressed as a percentage). For example, if a population grows by 7% per year, the doubling time is:
  • Doubling Time = 70 / 7 = 10 years
In the given exercise, the Earth's population grows at 1% per year. Using the formula, the doubling time is:
  • Doubling Time = 70 / 1 = 70 years
This formula stems from the rule of 70, a simplified version derived from more complex logarithmic calculations. It's an easy way to get quick approximations for exponential growth.
Population Growth Rate
The population growth rate is another key factor in studying demographics and exponential increases. It shows how rapidly a population is expanding over a year, usually as a percentage. For instance, in the exercise, the population grows at 1% annually. You can see the impact of different growth rates using the doubling time formula.
  • A higher rate (e.g., 7%) leads to a shorter doubling time (=10 years)
  • A lower rate (e.g., 1%) results in a longer doubling time (=70 years)
Understanding growth rate helps in predicting future population sizes and planning for resources. It becomes easier to comprehend the compounding effect over multiple years. The growth rate provides a base to calculate how many cycles (i.e., doubling) are needed to reach a specific multiple of the starting population.
Exponential Increase
Exponential increase describes growth in which the rate of change is proportional to the current value. This makes the population size increase faster as it grows larger. Here's how it works: If a population doubles every 70 years at 1% growth, in 140 years (2 x 70), the population will be four times the original size because each doubling cycle compounds on the previous one.
  • First 70 years: Doubles once (Original population x 2)
  • Next 70 years: Doubles again (2 x 2 = 4 times the original)
This idea can be mind-boggling and demonstrates the power of exponential functions. Just remember, exponential growth often appears rapid and unsustainable in real-world scenarios. Keeping this in mind helps contextualize future predictions and understand potential limits.

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Most popular questions from this chapter

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