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Chapter 1: Problem 12

What is the doubling time associated with \(3.5 \%\) annual growth?

Short Answer

Expert verified
Answer: Approximately 20 years.

Step by step solution

01

Convert the percentage growth rate to decimals

To use the Rule of 70, it is easier to work with decimals rather than percentages. To convert \(3.5 \%\) to a decimal, divide by \(100\). \(3.5\% = \frac{3.5}{100} = 0.035\)
02

Apply the Rule of 70

Insert our decimal growth rate into the Rule of 70 formula, which is Doubing Time = \(\frac{70}{\text{Growth Rate}}\), to find the doubling time: Doubling Time = \(\frac{70}{0.035}\)
03

Calculate the answer

Divide the numerator by the denominator to obtain the doubling time: Doubling Time = \(\frac{70}{0.035} = 2000\) The doubling time associated with \(3.5\%\) annual growth is approximately 20 years.

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Most popular questions from this chapter

In a classic story, a king is asked to offer a payment as follows: place one grain of rice on one square of a chess board ( 64 squares), then two on the next square, four on the next, 8 on the next, and double the previous on each subsequent square. The king agrees, not comprehending exponential growth. But the final number (adding all the grains) is one less than \(2^{64}\). How many grains is this?

A one-liter jar would hold about \(10^{16}\) bacteria. Based on the number of doubling times in our 24 -hour experiment, show by calculation that our setup was woefully unrealistic: that even if we started with a single bacterium, we would have far more than \(10^{16}\) bacteria after 24 hours if doubling every 10 minutes.

If a human body having an outward surface area of \(1 \mathrm{~m}^{2}\) continued to put out \(100 \mathrm{~W}\) of metabolic power in the form of infrared radiation in the cold of space (naked; no sun), what would the equilibrium temperature be? Would this be comfortable (put in context)?

A more dramatic, if entirely unrealistic, version of the bacteria-jar story is having the population double every minute. Again, we start the jar with the right amount of bacteria so that the jar will be full 24 hours later, at midnight. At what time is the jar half full now?

In the more dramatic bacteria-jar scenario in which doubling happens every minute and reaches single-jar capacity at midnight, at what time will the colony have to cease expansion if an explorer finds three more equivalent jars in which they are allowed to expand without interruption/delay?
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