Problem 74 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. After 100 years, a population whose growth rate is \(3 \%\) will have three times as many people as a population whose growth rate is \(1 \%\)

Step-by-Step Solution

Answer

No, the statement does not make sense. The population growth is not directly proportional to the population size because of the nature of exponential growth, where each year's growth is added to the original population and following years' growth is based on the larger population.

Step 1: Understand the Statement

The statement is suggesting that a population growing at a rate of \(3\%\) would be three times larger than a population growing at a rate of \(1\%\) after the same time period. This implies a direct relationship between the population size and the growth rate. It is essential to check this statement using the formula for compound growth.

Step 2: Apply Exponential Growth Formula

Use the exponential growth formula, \(A = P(1 + r)^n\), where \(A\) is the final amount, \(P\) is the initial population, \(r\) is the growth rate, and \(n\) is the time period. Calculate the final population size for both growth rates. This will allow us to compare the growth rates effect on their final populations over a 100 year period.

Step 3: Compare the Final Population Sizes

After calculating the final population size for the \(1\%\) and \(3\%\) growth rates, compare them. Remember to keep the initial population (\(P\)) the same for both calculations for valid comparison. If the population with a \(3\%\) growth rate is indeed three times larger than the population with a \(1\%\) growth rate, then the original statement is correct. If not, it is inaccurate.

Step 4: Reason the Outcome

Regardless the outcome from step 3, one needs to provide reasoning for it. The formula used calculates exponential growth because each year's growth is added to the original population, and following years' growth is based on the larger population. Thus the growth is compounded annually. This logic explains why a \(1\%\) growth rate doesn't result in a population one third the size of a \(3\%\) growth rate population over time.