Question

What is doubling time? What would be the doubling time for a population whose annual growth rate is \(5 \%\) ?

Short answer

The doubling time is the time it takes for a population to double in size due to a constant growth rate. For a population with an annual growth rate of 5%, the doubling time can be calculated using the exponential growth formula and solving for the time (t). The result is approximately 14.21 years, meaning it takes about 14.21 years for the population to double in size.

Step-by-step solution

Write the Exponential Growth Formula

Exponential growth follows the formula: \(P(t) = P_0(1 + r)^t\), where: - P(t) is the population size at time t - P_0 is the initial population size - r is the growth rate per time period (in this case annual growth rate) - t is the time (in this case, years) Since we are looking for the doubling time, we need to find the value of t when \(P(t) = 2P_0\).

Substitute the growth rate into the equation

The annual growth rate is given as 5%, which in decimal form is 0.05. Substitute the growth rate into the exponential growth formula: \(P(t) = P_0(1 + 0.05)^t\)

Set the equation to find the doubling time

To find the doubling time, we need to set \(P(t) = 2P_0\). So we have: \(2P_0 = P_0(1 + 0.05)^t\)

Solve for the doubling time (t)

Now, we need to solve the above equation for t: Divide both sides by \(P_0\): \(2 = (1 + 0.05)^t\) Take the natural logarithm of both sides: \(\ln{2} = \ln{(1 + 0.05)^t}\) Apply the power rule of logarithms: \(\ln{2} = t \cdot \ln{(1 + 0.05)}\) Divide both sides by \(\ln{(1 + 0.05)}\): \(t = \dfrac{\ln{2}}{\ln{(1 + 0.05)}}\) Using a calculator to find the values: \(t \approx 14.21\)

Interpret the result

The doubling time for a population with an annual growth rate of 5% is approximately 14.21 years. This means it takes about 14.21 years for the population to double in size.

Key concepts

Exponential Growth Formula

One of the fundamental concepts in understanding population dynamics is the exponential growth formula. It represents how populations grow at a constant rate over time. The formula is given by
\(P(t) = P_0(1 + r)^t\),
where

• \(P(t)\) is the population size after time t,
• \(P_0\) is the initial population size,
• \(r\) is the growth rate per time period, and
• \(t\) is the time.

When populations grow at a steady rate, the size of the population at later times can be predicted by this formula. When solving for doubling time, we set t as the unknown variable and manipulate the equation to determine how long it takes for the population to reach twice its original size.

Annual Growth Rate

The annual growth rate is a pivotal component in calculating the change in the population over a year. It's generally expressed as a percentage. To use the annual growth rate in exponential growth calculations, it needs to be converted to a decimal form. For instance, a growth rate of 5% should be converted to 0.05.
In the exercise provided, the annual growth rate is crucial for determining the population doubling time. It affects how rapidly a population increases, and higher growth rates lead to shorter doubling times. To apply this to the formula, we add 1 to the decimal form of the growth rate and raise this sum to the power of the number of years, t.

Solving Exponential Equations

Solving exponential equations, such as finding the doubling time of a population, often involves isolating the exponential expression and then using logarithms to solve for the unknown variable. The steps to do this follow a logical sequence:

1. Simplify the equation if necessary,
2. Isolate the exponential term,
3. Take the logarithm of both sides of the equation,
4. Use the properties of logarithms to simplify,
5. Solve for the unknown variable.

In our case, after setting the equation to find when the population size doubles, we take the natural logarithm of both sides to isolate t. This step is the trick to turning an equation that seems unsolvable in its current state into something more manageable.

Natural Logarithm

The natural logarithm, denoted as \(\text{ln}\), is the logarithm to the base e, where e is an irrational and transcendental number approximately equal to 2.71828. It plays a central role in solving exponential equations because it can reverse the process of exponential growth—turning multiplication into addition.
During our problem-solving procedure, the natural logarithm transforms the exponent in the equation to the coefficient of the logarithm. This is due to the property known as the power rule of logarithms, which states that \(\text{ln}(a^b) = b \times \text{ln}(a)\). By employing this rule, we can find the doubling time t by dividing the natural logarithm of 2 by the natural logarithm of the growth factor, facilitating the calculation of our final answer.