Question

The population in a certain town is increasing linearly each year. The population at time t=3 is 1285 and at time t=8 is 2460, where t is the number of years after 1990. If P(t) is the population at time t, write the equation that correctly represents this situation.

Answer 1

To find the equation of the linear population growth, first calculate the slope (m) using the given points and then apply it to the point-slope form with one of the points to derive the final equation, P(t) = 235t + 580.

Step-by-step explanation:

The student asked for the equation representing the linear population increase in a town given two population values at different times.

Using the two given points (3, 1285) and (8, 2460), where the first number in each pair represents the year (t) and the second number represents the population (P(t)), we can find the slope (m) of the line:

m = (2460 - 1285) / (8 - 3) = 1175 / 5 = 235.

Next, we can use the point-slope form of the linear equation which is P(t) - P1 = m(t - t1), where (t1, P1) is a point on the line (we can use either of the given points). Let's use the first point (3, 1285):

P(t) - 1285 = 235(t - 3).

This simplifies to:

P(t) = 235t + 580.

Therefore, the equation representing the population growth in this town is P(t) = 235t + 580, where t is the number of years after 1990.