Question

A city currently has 31,000 residents and is adding new residents steadily at the rate of 1200 per year. If the proportion of residents that remain after t years is given by S(t) = 1/(t + 1), what is the population of the city 7 years from now?

Answer 1

Answer:

Population of the city after 7 years from now, P(7) = 6370

Given:

Initial Population, $P_{i} = 31000$

rate, r(t) = 1200 /yr

S(t) = [/tex]\frac{1}{1 + t}[/tex]

Step-by-step explanation:

Let the initial population be  $P_{i} = 31000$

The population after T years is given by the equation:

$P(T) = P_{i}S(T) + \int_{0}^{T}S(T - t)r(t) dt$          (1)

Thus, the population after 7 years from now is given by using eqn (1):

$P(7) = \frac{3100}{1 + 7} + 1200\int_{0}^{7}\frac{1}{8 - t} dt$

$P(7) = 3875 - 1200ln(8 - t)|_{0}^{7}$

$P(7) = 3875 - 1200ln(8 - t)|_{0}^{7}$

$P(7) = 3875 - 1200(ln(1) - ln(8))$

$P(7) = 3875 + 2495 = 6370$