Question

In 2010, the US Census Bureau estimated the population of Phoenix to be 400.9 thousand and Baltimore, Maryland, to be 642 thousand people. Since 2010, Phoenix's population has been increasing at approximately 1.36% per year, and Baltimore's has been decreasing at 0.52% each year, assuming that the growth and decay rates remain constant.

Let p(t) represent the population of Miami and b(t) represent the population of Baltimore, t years after 2010. Determine the exponential functions that model the population (in thousands) of both cities.
Use the models of both to predict the population of both cities in 2022.
How many years for Baltimore's population to be cut in half?
When will the population of both cities be the same?

Answer 1

To model the populations of Phoenix and Baltimore, exponential functions are used. Phoenix's population in 2022 is predicted to be approximately 479.3 thousand, while Baltimore's is about 590.8 thousand. Formulas are provided for finding the time it will take for Baltimore's population to halve and when the populations of both cities will be equal.

Step-by-step explanation:

In order to model the populations of Phoenix and Baltimore, we can use the exponential growth and decay formulas, respectively. The formulas are as follows, with t representing the number of years since 2010:

• For Phoenix: p(t) = 400.9 × (1 + 0.0136)^t
• For Baltimore: b(t) = 642 × (1 - 0.0052)^t

To predict the populations in 2022, we simply plug t = 12 into each function:

• Phoenix: p(12) = 400.9 × (1 + 0.0136)^12 ≈ 479.3 thousand
• Baltimore: b(12) = 642 × (1 - 0.0052)^12 ≈ 590.8 thousand

To find how many years it will take for Baltimore's population to be cut in half, we use the formula:

b(t) = 642 × (1 - 0.0052)^t = 321

Solving for t gives us the desired number of years.

To determine when the populations of both cities will be the same, we set p(t) = b(t) and solve for t.