Question

Overtaking City A has a current population of 500.000 people and grows at a rate of \(3 \% / \mathrm{yr}\). City \(\mathrm{B}\) has a current population of 300,000 and grows at a rate of \(5 \% / \mathrm{yr}\) a. When will the cities have the same population? b. Suppose City C has a current population of \(y_{0}<500,000\) and a growth rate of \(p>3 \% / \mathrm{yr}\). What is the relationship between \(y_{0}\) and \(p\) such that Cities \(A\) and \(C\) have the same population in 10 years?

Short answer

Answer: The cities A and B will have the same population after approximately 20.65 years. The relationship between y0 and p for cities A and C to have the same population in 10 years is \(y_0 = \frac{669753}{(1+\frac{p}{100})^{10}}\).

Step-by-step solution

Writing the population functions of City A and City B

Use the exponential growth function to write the population functions of City A and City B. For City A: \(P_A(t) = 500000 * (1+0.03)^t\) For City B: \(P_B(t) = 300000 * (1+0.05)^t\)

Equating the population functions to find the time when the cities have the same population

Set the population functions of City A and City B equal to each other and solve for t: \(500000 * (1+0.03)^t = 300000 * (1+0.05)^t\)

Solving for t by taking the logarithm of both sides

To solve for t, take the logarithm of both sides: \(\log(500000 * (1+0.03)^t) = \log(300000 * (1+0.05)^t)\) Using the logarithm properties, we get: \(t*\log(1+0.03) - t*\log(1+0.05) = \log(\frac{300000}{500000})\) Now, we can factor out t and find its value: \(t(\log(1+0.03) - \log(1+0.05)) = \log(\frac{3}{5})\) \(t = \frac{\log(\frac{3}{5})}{\log(1+0.03) - \log(1+0.05)}\) Use a calculator to find the numerical value for t: \(t \approx 20.65\) So, after approximately 20.65 years, the cities will have the same population.

Finding the population of City A after 10 years

Now, let's find the population of City A after 10 years by plugging 10 into its population function: \(P_A(10) = 500000 * (1+0.03)^{10} \approx 669753\)

Writing the population function of City C and equating it to the population of City A after 10 years

For City C, its population function is given by: \(P_C(t) = y_0 * (1+(\frac{p}{100}))^t\) After 10 years, the population of City C should be equal to the population of City A after 10 years: \(y_0 * (1+\frac{p}{100})^{10} = 669753\)

Finding the relationship between y0 and p

Now, we need to find the relationship between y0 and p, so rewrite the equation in terms of y0: \(y_0 = \frac{669753}{(1+\frac{p}{100})^{10}}\) This is the relationship between y0 and p such that Cities A and C have the same population in 10 years.

Key concepts

Exponential Growth Function

An exponential growth function models situations where a quantity grows at a rate proportional to its current value, a phenomenon commonly observed in populations, finances, and natural processes.
In the context of population growth, the function takes the form of:
\[P(t) = P_0 \times (1 + r)^t\] where:

• \(P(t)\) is the population at time \(t\)
• \(P_0\) is the initial population
• \(r\) is the growth rate (expressed as a decimal)
• \(t\) is the time in consistent units, like years or months.

The power of the exponential growth function lies in the compound growth effect, exemplifying how even a small growth rate can lead to a large increase in population over time. This is precisely what can make City B, with a higher growth rate, potentially surpass City A in population, despite starting with a smaller population base.

Logarithmic Properties

Logarithms are incredibly handy in dealing with exponential equations since they allow us to bring down exponents and thus solve for variables that were previously in the exponent's position. Certain properties of logarithms are particularly useful:

• The product rule: \(\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)\)
• The quotient rule: \(\text{log}_b\left(\frac{x}{y}\right) = \text{log}_b(x) - \text{log}_b(y)\)
• The power rule: \(\text{log}_b(x^y) = y \times \text{log}_b(x)\)

These properties enable us to simplify and manipulate the terms in an exponential equation, as we can see in the population equation for Cities A and B. By using the quotient rule, the difference in growth rates becomes more visible, and it turns into a manageable form that can be solved for \(t\), the time when the populations are equal.

Solving Exponential Equations

Solving exponential equations, such as the one to determine when two cities will have equal populations, requires isolating the variable of interest. This calls for transformation skills made possible by logarithms. The general steps are:

• Isolate the exponential expression, if needed.
• Take the logarithm of both sides of the equation.
• Use logarithmic properties to simplify.
• Solve for the variable.

In our case, after taking the logarithms of both sides and using the properties, we can isolate \(t\), which then gives us the exact point in time when the populations of City A and B match. A calculator is typically used to find the numerical value of \(t\), showing a real-world application of logarithms in demography.

Understanding how to manipulate and solve these equations is not only crucial for mathematicians but also for demographers, economists, and scientists who regularly work with models predicting growth over time.