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Chapter 6: Problem 3

Explain the meaning of doubling time.

Short Answer

Expert verified
Answer: Doubling time is the period required for a quantity to double in size/value at a constant growth rate. It is used to measure the exponential growth of a certain variable, such as population, investments, or bacterial cultures. The formula for doubling time is t = (ln(2))/(ln(1+r)), where r is the growth rate. It can be used to determine how long it takes for a population to double its size or an investment to double its value at a given growth rate.

Step by step solution

01

Definition of Doubling Time

Doubling time is the period required for a quantity to double in size/value at a constant growth rate. It is typically used to measure the exponential growth of a certain variable, such as population, investments or bacterial cultures.
02

Formula for Doubling Time

The formula for doubling time can be derived from the exponential growth equation: P(t) = P0 * (1+r)^t, where P(t) is the final quantity at time t, P0 is the initial quantity, r is the growth rate, and t is the time. To find the doubling time, we need to find the time t required for P(t) to be twice P0. So, 2*P0 = P0*(1+r)^t. Simplifying and solving for t, we get the doubling time formula: t = \frac{ln(2)}{ln(1+r)}
03

Example: Population Growth

Suppose we have a population that grows at a rate of 2% per year. We can use the doubling time formula to find out how long it will take for the population to double its size. Using the formula t = \frac{ln(2)}{ln(1+r)}, we can substitute r = 0.02 (2% as a decimal) and find the doubling time: t = \frac{ln(2)}{ln(1+0.02)} \approx 35 years. So, it takes approximately 35 years for the population to double in size at a 2% annual growth rate.
04

Example: Investment Growth

Let's say an investment grows at a 7% annual interest rate, compounded yearly. We can use the doubling time formula to determine how long it will take for the investment to double its value. Using the formula t = \frac{ln(2)}{ln(1+r)}, we can substitute r = 0.07 (7% as a decimal) and find the doubling time: t = \frac{ln(2)}{ln(1+0.07)} \approx 10.24 years. So, it takes approximately 10.24 years for the investment to double its value at a compound interest rate of 7% per year.

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