Final answer:
a) The differential equation for the city population is dP/dt = kP. b) The solution to the differential equation is P = Po * e^(kt), where Po is the initial population and k is a constant of proportionality. c) The ratio of the population to the initial population after 75 years is given by P/Po = e^((ln(2) / 50) * 75)
Step-by-step explanation:
a) The differential equation for the city population can be set up as follows:
dP/dt = kP
where dP/dt represents the rate of change of the population over time, P represents the population at any given time, and k is a constant of proportionality.
b) To solve the differential equation, we can use the formula for exponential growth:
P = Po * e^(kt)
Given that the initial population (Po) is 52,000 and the population doubles in 50 years, we can substitute these values into the equation:
52,000 * 2 = 52,000 * e^(k * 50)
Simplifying the equation, we get:
e^(k * 50) = 2
Taking the natural logarithm of both sides, we have:
k * 50 = ln(2)
Dividing both sides by 50, we find:
k = ln(2) / 50
Now we can substitute the value of k back into the equation for exponential growth:
P = 52,000 * e^((ln(2) / 50) * t)
c) To find the ratio of the population (P) to the initial population (Po) after 75 years, we substitute t = 75 into the equation:
P/Po = e^((ln(2) / 50) * 75)