Question

In 2000, what was the population if it grew exponentially and was 1,200 in 2001 and 1,500 in 2002? A) 900 B) 800 C) 1,000 D) 1,100

Answer 1

To determine the population in 2000, we use the exponential growth formula. Solving the equations with the given data reveals that the population in 2000 was indeed 1,000. Thus the correct option is C.

Step-by-step explanation:

Exponential growth is modeled by the formula
`\( P_t = P_0 * (1 + r)^t \)`, where:

-
`\( P_t \)` is the population at the time ( t ),

-
`\( P_0 \)` is the initial population,

- ( r ) is the growth rate, and

- ( t ) is the time in years.

In this scenario, we are given
`\( P_1 = 1,200 \)` in 2001 and
`\( P_2 = 1,500 \)` in 2002. To find the growth rate ( r ), we can use the formula
`\( r = \frac{{P_1 - P_0}}{{P_0}} \)`, where
`\( P_1 \)` is the population in the first year.

Let's denote the initial population in 2000 as
`\( P_0 \)`. For the given data:

`\[ r = \frac{{1,200 - P_0}}{{P_0}} \]`

Similarly, for the next year:

`\[ 1,500 = P_0 * (1 + r)^2 \]`

Solving these equations simultaneously, we can find
`\( P_0 \)`, the population in 2000. Once we have
`\( P_0 \)`, we can substitute it into the exponential growth formula to find the population in 2000.

After solving the equations, we get
`\( P_0 = 1,000 \)`, that the population in 2000 was 1,000. Therefore, the correct answer is C) 1,000.