A county planner is looking at trends in population growth for two towns, Hapville and Merton.

The population of Hapville in the year 2000 was 18,000. Since then, the population has increased at a steady rate of 500 people per year.
The population of Merton is increasing according to the equation y=12,000(1.04)^t , where y= the population in thousands and t = the number of years since the year 2000.
If these trends continue, how will the populations of the two towns compare in 2030?

A.
Hapville will have about 6,000 more people than Merton.

B.
Merton will have about 6,000 more people than Hapville.

C.
Hapville will have about 21,000 more people than Merton.

D.
Merton will have about 21,000 more people than Hapville.

1 answer

To find out how the populations of Hapville and Merton compare in 2030, we will calculate the populations of both towns in that year.

For Hapville:

  • The population in the year 2000 = 18,000.
  • The population increases by 500 people per year.
  • From 2000 to 2030 is 30 years.

The population of Hapville in 2030 can be calculated as follows:

\[ \text{Population in 2030} = 18,000 + (500 \times 30) = 18,000 + 15,000 = 33,000 \]

For Merton:

  • The population follows the equation \( y = 12,000(1.04)^t \), where \( y \) is in thousands and \( t \) is the number of years since 2000.
  • For 2030, \( t = 30 \).

Using the formula:

\[ y = 12,000(1.04)^{30} \]

We need to calculate \( (1.04)^{30} \):

\[ (1.04)^{30} \approx 3.243 \]

Now substitute back into the equation:

\[ y = 12,000 \times 3.243 \approx 38,916 \text{ (in thousands)} = 38,916 \]

Comparing populations:

  • Hapville population in 2030: 33,000
  • Merton population in 2030: 38,916

Now we find the difference:

\[ 38,916 - 33,000 \approx 5,916 \]

This means Merton will have approximately 5,916 more people than Hapville in 2030.

Thus, the closest answer choice is:

B. Merton will have about 6,000 more people than Hapville.