Question

A city's population, P (in thousands), can be modeled by the equation P = 130(1.03)^x, where x is the number of years after January 1, 2000. For what value of r does the model predict that the population of the city will be approximately 170,000?

a) 9
b) 10
c) 11
d) 8

Answer 1

To find the value of x where the population model P = 130(1.03)^x predicts a population of approximately 170,000, we solve the equation 170 = 130(1.03)^x. Upon calculation, we find that x ≈ 9, meaning the population is estimated to be 170,000 in the year 2009.

Step-by-step explanation:

The student's question asks for the value of x for which the given population model predicts that the population will be approximately 170,000. The model is given by P = 130(1.03)^x, where P is the population in thousands, and x is the number of years after January 1, 2000. We want to solve for x when P equals 170 (note that 170,000 in the question refers to thousands).

First, we set up the equation: 170 = 130(1.03)^x. Next, we divide both sides by 130 to get 170/130 = (1.03)^x which simplifies to 1.3077 = (1.03)^x. We then take the natural logarithm of both sides to solve for x: ln(1.3077) = x ⋅ ln(1.03). Solving for x, we get x ≈ 9.

Thus, the model predicts that the population of the city will be approximately 170,000 after 9 years, which corresponds to the year 2009.