Problem 35 - Algebra and Trigonometry

Question

Use the formula \(t=\frac{\ln 2}{k}\) that gives the time for a population with a growth rate \(k\) to double to solve Exercises \(35-36 .\) Express each answer to the nearest whole year. The growth model \(A=4.3 e^{0.01 t}\) describes New Zealand's population, \(A,\) in millions, \(t\) years after 2010 . a. What is New Zealand's growth rate? b. How long will it take New Zealand to double its population?

Step-by-Step Solution

Answer

New Zealand's population growth rate is \(1\%\) per year, and it will take approximately 70 years for the population to double.

Step 1: Find the growth rate

To find the growth rate, we refer to the given exponential growth model \(A=4.3 e^{0.01 t}\). In this model, \(k=0.01\) represents the population's growth rate. Therefore, the population growth rate for New Zealand is \(0.01\) or, equivalently, \(1\% \) per year.

Step 2: Calculate the time required for population to double

The doubling time of a population can be calculated using the formula \(t=\frac{\ln 2}{k}\), where \(t\) is the doubling time in years, \(\ln 2\) is the natural logarithm of 2, and \(k\) is the growth rate. Since we found that \(k=0.01\), we substitute this value into our formula to solve for \(t\): \(t=\frac{\ln 2}{0.01}\) to get \(t \approx 70 \) years.