Problem 36 - College Algebra

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 \(A=112.5 e^{0.012}\) 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

Mexico's population growth rate is 0.012. It will take approximately 58 years for Mexico to double its population.

Step 1: Identify the Growth Rate

The given model for Mexico's population follows an exponential growth model \(A = 112.5e^{0.012}\). The growth rate \(k\) is the coefficient of \(t\) in the exponent of \(e\). Here, the rate of growth \(k\) is \(0.012\).

Step 2: Calculate the Time to Double the Population

Use the provided formula \(t=\frac{\ln 2}{k}\) to compute the time it will take for Mexico's population to double. Here, \(k\) is the growth rate found in the previous step (0.012). So the time is \(t=\frac{\ln 2}{0.012}\)

Step 3: Compute and Round Off

Calculate the value of \(t\). Upon calculation, you'll get t as approximately 57.7563 years. Since the question asks for rounding off the answer to the nearest whole year, the answer will be approximately 58 years.

Key Concepts

Population Doubling Time

Population doubling time refers to the amount of time it takes for a population experiencing exponential growth to double in size. This concept is particularly useful in demography, ecology, and economics, where understanding how quickly a population grows can inform policy and planning decisions. To find the doubling time, we use a special formula derived from the exponential growth equation. The formula is: \[ t = \frac{\ln(2)}{k} \] Here, \( t \) is the doubling time and \( k \) is the growth rate. The natural logarithm of 2 (\( \ln(2) \)) is approximately 0.693. This constant appears because doubling means increasing a quantity by a factor of two. By dividing it by the growth rate, you can calculate how long it takes the population to double. This concept helps us quantify growth in a simple and intuitive way.

Growth Rate Calculation

Calculating the growth rate is an essential step when using exponential equations to study trends over time. The growth rate \( k \) is a constant that shows how much a population increases per unit time. In the exponential growth formula expressed as \( A = A_0 e^{kt} \), it transforms the equation based on the time variable \( t \). When analyzing provided data, the growth rate can be identified easily in continuous compounding contexts or derived from given historical data trends. In our example, the model was \( A = 112.5e^{0.012t} \). The growth rate is directly visible in the exponential term, specifically \( k = 0.012 \). Knowing this value is paramount for further calculations like determining the time it takes for significant changes (e.g., population doubling), because it directly affects the timeline of changes. Hence, having an accurate growth rate ensures correct predictions about future population sizes.

Natural Logarithm Application

The natural logarithm, represented as \( \ln \), is a mathematical function that comes up often in the analysis of exponential growth problems. It is the inverse operation of the exponential function involving the constant \( e \), where \( e \approx 2.718 \). In our context, \( \ln \) is particularly useful in solving equations where the exponential is the subject. A common scenario is finding the doubling time of a population. The formula \( t = \frac{\ln(2)}{k} \) uses the natural logarithm because it simplifies the process of finding the time when a population doubles. Here, \( \ln(2) \) is used because doubling equates to a population factor increase of two. For descriptive purposes, the natural logarithm allows us to linearize growth processes, making intricate growth patterns more understandable and approachable. In computational terms, it allows us to "unwrap" exponential growth equations, making it easier to isolate variables like time or growth rate.