Problem 33 - College Algebra

Question

Use the exponential growth model, \(A=A_{0} e^{t_{i}},\) to show that the time it takes a population to double (to grow from \(A_{0}\) to \(2 A_{0}\) ) is given by \(t=\frac{\ln 2}{k}\)..

Step-by-Step Solution

Answer

The time it takes for a population to double in an exponential growth model is given by \(t=\frac{\ln 2}{k}\), where \(k\) is the growth rate.

Step 1: Insert the Doubling Population

Firstly, understand that for the population to double, \(A\) would become \(2A_{0}\). Substitute \(2A_{0}\) into the equation for \(A\): \(2A_{0}=A_{0} e^{kt}\).

Step 2: Simplify the equation

Next, simplify the equation by dividing both sides by \(A_{0}\) to isolate the exponential term: \(2=e^{kt}\).

Step 3: Apply Natural Logarithm

Then, apply the natural logarithm (ln) to both sides of the equation to get rid of \(e\) on the right side. The natural logarithm of \(e^x\) equals \(x\). So we get, \(\ln 2 = \ln e^{kt}\). By the properties of logarithms, we know that \(\ln e^{kt} = kt\). So, \(\ln 2 = kt\).

Step 4: Solve for Doubling Time

Finally, solve for the variable \(t\) (which represents the doubling time of the population) by dividing the equation by \(k\). The equation becomes: \(t=\frac{\ln 2}{k}\).

Key Concepts

Population Doubling Time

When studying population growth, one significant concept is the 'population doubling time', which refers to the period it takes for a given population to double in size. This measure is pivotal in fields like biology, demography, and environmental science, as it provides valuable insight into the rate at which species, including humans, proliferate.

In the context of the exponential growth model, assuming that a population grows at a constant rate, the doubling time can be derived mathematically. The formula for population doubling time is captured by the equation: \[ t = \frac{\ln 2}{k} \. \]Here, \( t \) is the doubling time, \( \ln 2 \) represents the natural logarithm of 2 (which is a constant approximately equal to 0.693), and \( k \) is the growth rate constant. By applying this equation, we can estimate how quickly a population will grow under constant conditions, which is crucial for planning and resource allocation in various sectors.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a mathematical operation that is intrinsically related to the special number 'e', approximately equal to 2.71828. This number 'e' is the base of the natural logarithm and it holds a unique place in mathematics due to its continuous growth properties.

The natural logarithm of a number is the exponent to which 'e' must be raised to produce that number. So when you see an expression like \( \ln x \), it asks the question: 'To what power must we raise 'e' to get \( x \)?'. The function \( \ln \) is the inverse operation to exponentiating with base 'e'. For example, if we have \( e^y = x \), then it follows that \( \ln x = y \).

One practical application of the natural logarithm is when solving exponential equations. By taking the natural logarithm of both sides of an equation, you can 'unfold' the exponential function, making it easier to isolate variables and solve the equation, as seen in the steps to find population doubling time.

Exponential Equations

Exponential equations are mathematical expressions where variables appear as exponents, such as \( A = A_0 e^{kt} \. \) They depict situations where a quantity grows or decays at a rate that is proportional to its current value, which makes them fitting for modeling processes like population growth, radioactive decay, and interest compounding.

The exponential growth model typically contains a base 'e', which represents the natural exponential function. To solve exponential equations, we often use logarithms since they allow us to bring the exponent down to where we can handle it algebraically. In the context of our original problem, we used a natural log to solve for the doubling time of a population.

It's essential to master the manipulation of exponential equations as they frequently occur across many scientific and mathematical disciplines. With a solid understanding of how to use natural logarithms to solve these equations, one can tackle a wide range of real-world problems.