SOLUTION: The population of a small town has been gradually decreasing over the past number of years. The equation: A = 1236(.97)t, where t is time in years, models the decrease in the town

Question 277891: The population of a small town has been gradually decreasing over the past number of years. The equation: A = 1236(.97)t, where t is time in years, models the decrease in the town�s population.
In how many years will it take for the population to reach 846 people? (Round to the nearest hundredth of a year.)

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
your formula should read:

A = 1236 * (.97)^T where:

A is the future population.
T is the time in years.

The ^ indicates exponentiation.

According to the equation model, the initial population is 1236.

The general form of the equation is:

F = P * (1+G)^T where:

F = the future population
P = the present population
G = the annual growth rate
T = the number of years

In your equation:
F = A
P = 1236
G = -.03 because 1 - .03 = .97
T = number of years

Your equation becomes:

A = 1236 * .97^T

Since A is the future population, then A becomes 846.

Your equation becomes:

846 = 1236 * .97^T

Divide both sides of this equation by 1236 to get:

846/1236 = .97^T

Take the log of both sides of this equation to get:

log(846/1236) = log(.97^T)

Since log(a^b) = b*log(a), your equation becomes:

log(846/1236) = T*log(.97)

Divide both sides of this equation by log(.97) to get:

log(846/1236)/log(.97) = T which is the same as:

T = log(846/1236)/log(.97)

use your calculator to find the logs which make your equation equal to:

T = -.164648108 / -.013228266 = 12.44668886

The future population will be 846 in 12.4466886 years.

1236 * .97^(12.44668886) = 846

Round that to the nearest hundredth of a year and you get:

T = 12.45

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