Question

In 1980, the population of a town was 2000 people. The exponential function P(t)=2000(1.05)t models the population of the town since 1980. Which statements are true regarding the population of the town? Select THREE that apply.

A.

The population has increased by 5% each year since 1980.

B.

The population has increased by 105% each year since 1980.

C.

The population has increased by 5 people each year since 1980.

D.

Each year since 1980, the population is 105% of the population of the previous year.

E.

Each year since 1980, the population is 1.05 times the population of the previous year.

Answer 1

The population of the town increases by 5% each year, is 105% of the population of the previous year, and the population is 1.05 times the population of the previous year. Therefore, statements A, D, and E are true.

Step-by-step explanation:

The exponential function P(t)=2000(1.05)^t models the population growth of a town since 1980. To understand which statements are true regarding the population of the town, we need to interpret the parts of the function:

The base number 2000 represents the initial population of the town in 1980.

The number 1.05 in the equation is known as the growth factor.

The exponent t represents time in years since 1980.

Given these interpretations, let's examine which statements are correct:

A: The population has increased by 5% each year since 1980. This is true because 1.05 represents a 5% increase when comparing each year to the previous one.

C: The population has increased by 5 people each year since 1980. This is false because the growth is exponential, not linear.

D: Each year since 1980, the population is 105% of the population of the previous year. This is true because the growth factor of 1.05 implies this exact scenario. This is just a restatement of option A but in a different form.

E: Each year since 1980, the population is 1.05 times the population of the previous year. This is true and it conveys the same information as D. It is the mathematical expression of the growth factor's impact on the population, indicating an increase by the factor of 1.05 year over year.

Therefore, the correct answers are A, D, and E.

Answer 2

B. The population has increased by 105% each year since 1980.

D. Each year since 1980, the population is 105% of the population of the previous year.

E. Each year since 1980, the population is 1.05 times the population of the previous year.

Step-by-step explanation:

From the question, we are given the exponential function:

The exponential function P(t)=2000(1.05)^t models the population of the town since 1980.

Exponential growth rate =

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

Where r = growth rate in percentage

P(t) = Size of the population after t years

t = time in years

Hence,

From the formula above, we can say =

1 + r = 1.05

r = 1.05 - 1

r = 0.05

Converting to percentage = 0.05 × 100 = 5%

1.05 can also converted to percentage and this = 1.05 × 100 = 105 %

The population has increased by 105% each year since 1980.

Hence, Options, B, D and E is correct