Question

The population of Centerville increases each year. The function C(t) = P(1 +r)^t represents the population of centerville at year t, where P is the initial population and r is the rate of increase. Select the correct statement.
A. The factor (1 + r) is the percentage by which the population grows each year.

B. The value of C depends only on starting population and year.

C. r is less than 0.

D. P represents the population in year 0

Answer 1

Answer:

Option D) P represents the population in year 0

Step-by-step explanation:

We are given the following information in the question:

$C(t) = P(1 +r)^t$

where C(t) represents the population of Centerville at year t, where P is the initial population and r is the rate of increase.

If we put t = 0 in the given function, we have,

$c(0) = P(1+r)^0 = P$

Hence, it could be said  P represents the population in year 0 or P is the initial population.

Hence, Option D) is the correct statement.

The correct statements are:

A) r is the rate by which the population increases

B)  The value of C depends on starting population, r the rate of increase of population and year.

C) r cannot be less than zero

Answer 2

Answer:

P represents the population in year 0 ⇒ D

Step-by-step explanation:

* Lets explain the exponential growth function

- The exponential growth function is f(x) = a (1 + r)^t, where a is the initial

  amount (at t = 0), (1 + r) is the factor of growth , r is the rate of growth

  in decimal ant is the time of growth

* Lets solve the problem

∵ The function C(t) = P(1 + r)^t represents the population of

  centerville at year t, where P is the initial population and r is the

  rate of increase

- Ex: If your investment is increased 10% annually, then that means

 each year, your total has multiplied itself by 110% (the growth factor

 is 1 + 10/100 = 1.1)

∴ (1 + r) is the factor grows each year

∵ C(t) = P(1 + r)^t

∴ C depends on P(starting population) , r(the increasing rate and

   t(the time in year)

∵ r is the rate of increase means the percentage of increasing , then

  0 < r < 1

∴ r is not less than 0

∵ P is the initial amount when t = 0

∴ P represents the population in year 0