Problem 6 - College Algebra

Question

The exponential models describe the population of the indicated country, \(A,\) in millions, \(t\) years after \(2010 .\) Use these models to solve Exercises \(1-6\) Indis \(\quad A=1173.1 e^{0.005 t}\) Ing \(\quad A=31.5 e^{0.018}\) Japer \(\quad A=127.3 e^{-0.006 t}\) Restis \(\quad A=141.9 e^{-0.005 t}\) When will India's population be 1491 million?

Step-by-Step Solution

Answer

The value of \(t\) obtained in step 6 is the year, after 2010, when the population of India will be 1491 million.

Step 1: Substitute the given values into the equation

Start by substituting \(A = 1491\) into the equation given for the population in India. This results in the following equation: \(1491 = 1173.1 e^{0.005 t}\)

Step 2: Isolate the exponential term

To isolate the exponential term \(e^{0.005 t}\), divide both sides of the equation by \(1173.1\). This gives us: \(\frac{1491}{1173.1} = e^{0.005 t}\).

Step 3: Simplify the left side

Simplify the left hand side of the equation gives: \(1.2708 = e^{0.005 t}\)

Step 4: Apply the Natural Logarithm

Apply the natural logarithm (ln) to both sides, which will enable us to eliminate the exponential \(e\). The rule \(\ln(e^x) = x\) applies here, resulting in: \(\ln(1.2708) = 0.005 t\)

Step 5: Isolate the variable \(t\)

Finally, solve for \(t\) by dividing both sides of the equation by \(0.005\), yielding \(t = \frac{\ln(1.2708)}{0.005}\). The result will be the number of years after 2010 in which India's population will be 1491 million.

Step 6: Calculate \(t\)

Using a calculator, compute the right-hand side to get a numerical value of \(t\)

Key Concepts

Exponential Growth and Decay

Exponential growth and decay are mathematical concepts used to describe the increase or decrease of a quantity over time. This phenomenon is commonly observed in situations like population growth, radioactive decay, and interest accumulation.

An exponential function has the general form of \(A = P e^{rt}\), where \(A\) is the ending amount, \(P\) is the initial amount, \(r\) is the rate of growth or decay, \(t\) is time, and \(e\) is the base of natural logarithms, approximately 2.71828. If \(r\) is positive, the function represents exponential growth, and if \(r\) is negative, it illustrates exponential decay.
In the context of population models, a positive rate indicates that the population is increasing over time, whereas a negative rate signifies a declining population.

Natural Logarithms

Natural logarithms are a specific type of logarithm with the base \(e\), which is an irrational and transcendental number approximately equal to 2.71828. The notation used for natural logarithms is \(\ln\).

One of the key properties of natural logarithms is the identity \(\ln(e^x) = x\), which provides a powerful tool for solving exponential equations. This identity means that taking the natural logarithm of \(e\) raised to a power will simply yield that power. For example, \(\ln(e^3) = 3\).

Solving Exponential Equations

To solve exponential equations, you need to isolate the variable that’s in an exponent. For equations in the form of \(A = P e^{rt}\), where \(A\), \(P\), and \(r\) are known, and \(t\) is the variable, the following steps are typically employed:

• Isolate the exponential term \(e^{rt}\) by dividing both sides by \(P\).
• Apply the natural logarithm to both sides of the equation to remove the base \(e\), using the property that \(\ln(e^x) = x\).
• Solve for the variable \(t\) by dividing both sides by \(r\) if necessary.

These steps allow one to solve for the unknown time \(t\), which can be difficult to grasp without using logarithms due to the exponential nature of the equation.

Population Growth Analysis

Population growth analysis involves studying the changes in population sizes over time and is vital for resource planning, environmental policy, and understanding social dynamics. It uses mathematical models, such as the exponential model, to predict future population sizes based on current data and growth rates.

The exponential growth formula \(A = P e^{rt}\) is often used to model populations where \(A\) is the future population size, \(P\) is the current population size, \(r\) is the growth rate, and \(t\) is the time in years into the future. By manipulating this formula and employing natural logarithms for solving, demographers and statisticians can estimate when a population will reach a certain size, assuming no significant changes in growth rate.