Question

Use \(y=y_{0} e^{k t}\). The populations of New York and Los Angeles are growing at \(1 \%\) and \(1.4 \%\) a year, respectively. Starting from 8 million (New York) and 6 million (Los Angeles), when are the populations equal?

Short answer

The populations become equal in approximately 78 years.

Step-by-step solution

Write the Population Formulas

For New York, the formula is \(y_{NY} = 8 \, e^{0.01t}\). For Los Angeles, the formula is \(y_{LA} = 6 \, e^{0.014t}\). These represent exponential growth with the respective percentages as growth rates.

Set the Equations Equal

We need to find when the populations are equal, so we set the two equations equal to each other: \(8 \, e^{0.01t} = 6 \, e^{0.014t}\).

Simplify the Equation

Divide both sides by 6 to isolate terms with \(t\): \(\frac{8}{6} = e^{0.014t - 0.01t}\), which simplifies to \(\frac{4}{3} = e^{0.004t}\).

Take the Natural Logarithm

Take the natural logarithm of both sides to solve for \(t\): \(\ln\left(\frac{4}{3}\right) = 0.004t\).

Solve for Time \(t\)

Divide both sides by 0.004 to solve for \(t\): \(t = \frac{\ln\left(\frac{4}{3}\right)}{0.004} \approx 77.66\).

Key concepts

Population Growth Modeling

Population growth modeling is a way to predict how populations change over time. In mathematics, this often involves creating formulas that can describe the increase in population size. Here, we use exponential growth models, which are useful when the rate of population change is proportional to the current population.

• Populations can grow without bound, limited only by environmental factors.
• Exponential models help in predicting long-term outcomes for large cities.
• In our example, New York and Los Angeles are modeled with different annual growth rates, leading to predictions about when these populations will equalize.

By using exponential growth models, we can predict that New York and Los Angeles' populations will equal each other, given their respective growth rates and starting sizes.

Exponential Equations

Exponential equations are a type of mathematical expression where variables are raised to a power. These are crucial in modeling biological, physical, and financial processes where growth occurs swiftly and exponentially. Here, the equations take the form of New York's and Los Angeles' population formulas:

• For New York: \(y_{NY} = 8 \, e^{0.01t}\)
• For Los Angeles: \(y_{LA} = 6 \, e^{0.014t}\)

In these equations, \(e\) is the base of the natural logarithm, and the exponent is the product of the growth rate and time. Setting these two equations equal allows us to determine when the two populations will be the same:

• The simplified form: \(\frac{4}{3} = e^{0.004t}\)

These equations reveal when two processes that grow at different rates meet at a common point through careful manipulation and solving of exponents.

Natural Logarithms

Natural logarithms are used to solve exponential equations. They help in "undoing" the exponentiation process, allowing us to solve for time or other variables in exponential equations. In the case of our population equations, we take the natural logarithm of both sides:

• Inequality before logarithm: \(\frac{4}{3} = e^{0.004t}\)
• Apply natural logarithm: \(\ln\left(\frac{4}{3}\right) = 0.004t\)

Once the natural log is applied, it simplifies the problem—turning multiplication into addition and dealing with exponents more manageably. The final step is simply to divide by the exponential coefficient to isolate and solve for time:

• Solve for \(t\): \(t = \frac{\ln\left(\frac{4}{3}\right)}{0.004} \approx 77.66\)

Natural logarithms provide the mathematical tools to transform complex exponential equations into linear ones that are much easier to solve.