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

Option D) P represents the population in year 0

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

P represents the population in year 0 ⇒ D

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