Problem 93
Question
The Tarheel State. The \(4.3 \%\) annual population growth rate for the Raleigh- Cary metropolitan area in North Carolina is one of the largest of any metropolitan area in the United States. If its growth rate remains constant, how long will it take for its population to double?
Step-by-Step Solution
Verified Answer
It will take approximately 16.28 years for the population to double.
1Step 1: Understand the problem
We need to find out the time required for the population to double given a consistent annual growth rate. The formula to determine the doubling time in this context is related to the Rule of 70.
2Step 2: Apply the Rule of 70
The Rule of 70 is a way to estimate the number of years required to double the population given a constant annual growth rate. It states that Doubling Time (in years) = \( \frac{70}{\text{Annual Growth Rate (%)}} \).
3Step 3: Calculate Using the Given Growth Rate
Substitute the given population growth rate into the formula: \( \text{Doubling Time} = \frac{70}{4.3} \).
4Step 4: Perform the Calculation
Calculate \( \frac{70}{4.3} \approx 16.28 \). This means it will take approximately 16.28 years for the population to double given a 4.3% annual growth rate.
Key Concepts
Rule of 70Doubling TimeExponential Growth
Rule of 70
The Rule of 70 is a handy formula used in mathematics and economics to calculate the doubling time of a population or investment, based on a constant annual growth rate. This rule offers a quick and simple way to understand how long it will take for a quantity growing at a certain rate to double in size. This estimation is straightforward and doesn't require complex calculations.
To use the Rule of 70, simply divide 70 by the annual growth rate percentage. The result will give you the approximate number of years it will take for the population to double in size. For example, with an annual growth rate of 4.3%, the doubling time can be calculated as \( \frac{70}{4.3} \approx 16.28 \) years.
The genius of the Rule of 70 lies in its simplicity. Though it's an estimation, it provides a surprisingly accurate figure that helps in both academic studies and real-world applications. The reason this calculation works is because of the mathematical relationship in exponential growth between the natural logarithm and the constant rate of growth.
To use the Rule of 70, simply divide 70 by the annual growth rate percentage. The result will give you the approximate number of years it will take for the population to double in size. For example, with an annual growth rate of 4.3%, the doubling time can be calculated as \( \frac{70}{4.3} \approx 16.28 \) years.
The genius of the Rule of 70 lies in its simplicity. Though it's an estimation, it provides a surprisingly accurate figure that helps in both academic studies and real-world applications. The reason this calculation works is because of the mathematical relationship in exponential growth between the natural logarithm and the constant rate of growth.
Doubling Time
Doubling time is the period it takes for a quantity that grows exponentially to double in size, whether it be a population or investment. This concept is particularly important in understanding how quickly something can grow over a period of time when it is subjected to a constant growth rate. The concept is rooted in exponential growth, a powerful model that describes how populations or investments can expand over time.
To determine the doubling time, one can apply the Rule of 70. You take the number 70 and divide it by the annual percentage growth rate. For example, with our constant annual growth rate of 4.3%, the doubling time is: \( \frac{70}{4.3} \approx 16.28 \) years.
To determine the doubling time, one can apply the Rule of 70. You take the number 70 and divide it by the annual percentage growth rate. For example, with our constant annual growth rate of 4.3%, the doubling time is: \( \frac{70}{4.3} \approx 16.28 \) years.
- The calculated doubling time is useful for projections.
- It helps planners and economists make forecasts about future population sizes or investment returns.
- This concept is crucial for understanding limits to growth and planning for sustainable development.
Exponential Growth
Exponential growth occurs when the growth rate of a mathematical function is directly proportional to the current size, meaning that as the quantity grows, it will grow faster and faster. This type of growth is often visualized as an upward-sloping curve that becomes steeper over time.
In the context of population growth, it means that the more people there are, the more individuals there are to continue the growth, which can lead to rapid increases in numbers over time if the rate remains constant. It is this property of exponential growth that makes the Rule of 70 and doubling time interesting and significant.
In the context of population growth, it means that the more people there are, the more individuals there are to continue the growth, which can lead to rapid increases in numbers over time if the rate remains constant. It is this property of exponential growth that makes the Rule of 70 and doubling time interesting and significant.
- Exponential growth models are used in various fields, from biology to finance.
- This model helps to predict future growth patterns.
- It is fundamental for understanding long-term trends and outcomes in systems where growth is compounded.
Other exercises in this chapter
Problem 91
Use a calculator to solve each equation. Round answers to four decimal places. See Example 6. $$ \log x=3.7813 $$
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Look Alikes a. \(\log _{2}(x+5)-\log _{2} 4 x=\log _{2} x\) b. \(\ln (x+5)-\ln 4 x=\ln x\)
View solution Problem 93
Write out in words how to say each of the following: $$ (f \circ g)(2) \quad g(f(-8)) $$
View solution Problem 93
Look Alikes \(\cdots\) a. \(\log _{2}(x+5)-\log _{2} 4 x=\log _{2} x\) b. \(\ln (x+5)-\ln 4 x=\ln x\)
View solution