Problem 35 - College Algebra Essentials

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

a. New Zealand's growth rate is 0.01. b. It will take approximately 69 years for New Zealand to double its population.

Step 1: Identify the growth rate from the model

The model provided for New Zealand's population is given as \(A=4.3e^{0.01t}\). In this model, the growth rate \(k\) is represented by the coefficient of \(t\) in the exponent, which is 0.01. Hence, New Zealand's growth rate is 0.01.

Step 2: Substituting the growth rate into the doubling time formula

The formula provided for the time \(t\) it takes a population to double is \(t = \frac{\ln 2}{k}\). Substituting \(k=0.01\) into this formula gives \(t = \frac{\ln 2}{0.01}\).

Step 3: Evaluating the expression

The value of \( \ln 2 \) is approximately 0.693. Hence, substituting this into the evaluated expression would give us our answer. Hence, \(t = \frac{0.693}{0.01} = 69.\)

Step 4: Rounding to the nearest whole year

Although the exact result of the operation is 69.3, the exercise requests that the final result be rounded to the nearest whole year, which is 69 years when rounded down.