Problem 36 - 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 \(A=112.5 e^{0.012 t}\) describes Mexico's population, \(A,\) in millions, \(t\) years after 2010 . a. What is Mexico's growth rate? b. How long will it take Mexico to double its population?

Step-by-Step Solution

Answer

a. Mexico's annual growth rate is 1.2%. b. It will take Mexico approximately 58 years to double its population.

Step 1: Identify the Growth Rate

From the given model \(A=112.5 e^{0.012 t},\) it's clear to see that the exponential growth rate, \(k,\) is 0.012 as this is the value multiplied by \(t\) in the power of \(e,\) according to the general exponential growth formula \(A=A_0 e^{kt}.\) Therefore, Mexico's growth rate is 0.012 or 1.2% per year.

Step 2: Calculate the Time to Double Population

To calculate how long it will take for Mexico to double its population, we need to substitute the value of \(k\) that we already found into the formula \(t=\frac{\ln 2}{k}.\) This gives us \(t=\frac{\ln 2}{0.012}.\)

Step 3: Solve the Equation to Find Time

To solve for time, calculate the right-hand side of the equation. This will give the amount of time, \(t,\) it will take for Mexico to double its population. Round this time to the nearest whole year for the final answer.