Final answer:
To calculate the number of years after 2012 when the population reaches 15,000, divide both sides of the equation 15,000 = 10,000e^(kt) by 10,000 and then take the natural logarithm, resulting in t = ln(1.5) / k, where k is the growth rate.
Step-by-step explanation:
To find the number of years after 2012 that the population of a town will be 15,000 using the formula P = 10,000e^(kt), where P is the population, e is the base of the natural logarithm, k is the growth rate, and t is the time in years after 2012, you would need to solve the equation 15,000 = 10,000e^(kt).
The first step is to isolate the exponential term:
Divide both sides by 10,000, resulting in 1.5 = e^(kt).
Next, take the natural logarithm of both sides to solve for t as the natural logarithm function is the inverse of the exponential function:
Apply the natural logarithm: ln(1.5) = kt.
Divide by k to solve for t: t = ln(1.5) / k.
Note that the value of k (the growth rate) is not provided, hence it is necessary to know k to calculate the exact number of years.