Overtaking City A has a current population of 500,000 people and grows at a rate of 3%/yr. City B has a current population of 300,000 and grows at a rate of 5%/yr.
b. Suppose City C has a current population of y₀ < 500,000 and a growth rate of p > 3%/yr. What is the relationship between y₀ and p such that Cities A and C have the same population in 10 years?

Verified step by step guidance

1. Express the population of City A after 10 years using the exponential growth formula: \(P_A(10) = 500,000 \times (1 + 0.03)^{10}\), where 0.03 represents the 3% growth rate.
2. Express the population of City C after 10 years similarly: \(P_C(10) = y_0 \times (1 + p)^{10}\), where \(y_0\) is the initial population of City C and \(p\) is its growth rate (expressed as a decimal).
3. Set the populations equal to find when City A and City C have the same population after 10 years: \$500,000 \times (1 + 0.03)^{10} = y_0 \times (1 + p)^{10}$.
4. Isolate \(y_0\) to express it in terms of \(p\): \(y_0 = \frac{500,000 \times (1 + 0.03)^{10}}{(1 + p)^{10}}\).
5. Interpret this relationship: for City C to catch up with City A in 10 years, its initial population \(y_0\) and growth rate \(p\) must satisfy the equation above, with \(y_0 < 500,000\) and \(p > 0.03\).

Key Concepts

Exponential Growth Model

Population growth at a constant percentage rate is modeled by exponential functions, where the population at time t is given by P(t) = P₀ * (1 + r)^t. Here, P₀ is the initial population, r is the growth rate per time period, and t is the number of time periods. This model helps predict future populations based on current data.

Equating Populations to Find Conditions

To find when two populations are equal, set their exponential growth expressions equal and solve for the unknown variable. This involves forming an equation like P₀₁*(1 + r₁)^t = P₀₂*(1 + r₂)^t and manipulating it to express one variable in terms of others, which reveals the relationship between initial populations and growth rates.

Logarithmic Functions for Solving Exponential Equations

When solving equations involving variables in exponents, logarithms are used to isolate the variable. Taking the natural log or log base 10 of both sides allows the exponent to be brought down as a multiplier, enabling algebraic manipulation to find unknown growth rates or initial populations.