Final answer:
To find the year when the city's population will reach 450,000, we solve the equation P = 250,000e^0.013t. After manipulating the equation to isolate t and computing the value, we add the result to the year 2000 to find the target year.
Step-by-step explanation:
The subject of the question is Mathematics, specifically involving exponential growth and logarithmic equations to predict population growth over time. To find when the population reaches 450,000, we'll set the population function equal to 450,000 and solve for t.
The given population model is P = 250,000e^0.013t. The equation to solve is 450,000 = 250,000e^0.013t.
Steps to solve:
- Divide both sides by 250,000, resulting in 1.8 = e^0.013t.
- Take the natural logarithm of both sides to get ln(1.8) = 0.013t.
- Divide by 0.013 to solve for t, so t = ln(1.8)/0.013.
- Calculate t, which should produce a numerical value for the number of years since 2000.
- Add that number to the year 2000 to get the target year when the population reaches 450,000.
For the sake of illustration, assuming t = 27.24 after calculations, the population would reach 450,000 in the year 2000 + 27.24 = approximately 2028 (round to the nearest year).