Question

1. In 2023, cities X and Y are nearly the same size and both are growing

rapidly. City X has a population of 100,000 and is growing at a continuous yearly rate
of 2.6%. The population of city Y is 90,000 and is growing at a continuous yearly rate
of 3.1%. Find each of the following.

(a) Assuming continued growth at the same rate, find a formula for the
population of city X at time t years after 2023.

(b) Assuming continued growth at the same rate, find a formula for the
population of city Y at time t years after 2023.

(c) Write and solve an equation that can be used to determine the number of
years it will take for the population of city Y to catch up to city X. Show your work pretty please!!!!

Answer 1

Explanation:

(a) To find the population of city X at time t years after 2023, we can use the formula:

P(t) = P(0) * e^(rt)

where P(0) is the initial population, r is the annual growth rate as a decimal, and e is the mathematical constant approximately equal to 2.71828.

Substituting the values given, we get:

P(t) = 100,000 * e^(0.026t)

(b) Similarly, for city Y, we have:

P(t) = 90,000 * e^(0.031t)

(c) To find the number of years it will take for the population of city Y to catch up to city X, we can set the two population formulas equal to each other and solve for t:

100,000 * e^(0.026t) = 90,000 * e^(0.031t)

Dividing both sides by 90,000 and taking the natural logarithm of both sides, we get:

ln(e^(0.026t)) = ln(e^(0.031t)) + ln(9/10)

0.026t = 0.031t + ln(9/10)

0.005t = ln(9/10)

t = ln(9/10) / 0.005

t ≈ 13.9 years

Therefore, it will take approximately 13.9 years for the population of city Y to catch up to city X.