Problem 74 - Algebra and Trigonometry

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

The statement does not make sense. The comparison of population sizes given the compounded growth over time does not result in a simple linear relationship (e.g., 3% growth resulting in three times the population of 1% growth). Exponential growth results in a much more significant population size as the growth rate increases.

Step 1: Set-up the Population Growth Formula

Use the exponential growth formula to calculate the final population after 100 years for both growth rates. Set \(P_0\) (initial population) as \(x\) for simplicity.

Step 2: Calculate the Population for a 3% Growth Rate

Substitute \(r = 0.03\) and \(t = 100\) into the exponential growth formula. Simplify to get \(P = x e^{0.03*100}\).

Step 3: Calculate the Population for a 1% Growth Rate

Substitute \(r = 0.01\) and \(t = 100\) into the exponential growth formula. Simplify to get \(P = x e^{0.01*100}\).

Step 4: Compare the Two Calculated Populations

At this point, compare the populations obtained in steps 2 and 3. Is the population for a 3% growth rate three times greater than that for a 1% growth rate?