Final answer:
To derive the formula for the population size following an exponential function, we utilize the exponential growth function P(t) = P_0 * e^(rt) with given population data for two points in time and solve for the initial population size P_0 and growth rate r.
Step-by-step explanation:
To find the formula for the population size that follows an exponential function, we can use the general form of an exponential growth function, P(t) = P_0 * e^(rt), where P(t) is the population at time t, P_0 is the initial population size, r is the growth rate, and e is the base of the natural logarithm.
Given the population size of a city was 79,000 in 1990 and 136,000 in 2005, we can set up two equations based on the exponential model:
- P(1990) = 79,000 = P_0 * e^(r*1990)
- P(2005) = 136,000 = P_0 * e^(r*2005)
We can then solve these equations simultaneously to find the values of P_0 and r. From there, we can derive the exact formula for the population growth.
However, additional steps involving logarithms and algebra are needed to find the exact values of P_0 and r, which we haven't computed here.