Question

The population of a city with 15,000 people declines at a rate of 2% each year.

Which is an exponential equation that models the population, y, after x years?



y=15,000⋅(98)x
y is equal to 15 comma 000 times 98 to the x th power

y=15,000⋅0.98x
y is equal to 15 comma 000 times 0 point 9 8 x

y=15,000⋅(2)x
y is equal to 15 comma 000 times 2 to the x th power

y=15,000⋅(0.98)x

Answer 1

To model the population of a city that declines at a rate of 2% each year, we need to express the decline as a multiplier.

A decline in the population by 2% means that each year, 98% of the population remains. This can be represented as a multiplier of \(0.98\) (which is equal to \(1 - 0.02\)).

The general formula for exponential decay is:

\[
y = P \cdot (1 - r)^x
\]

where:
- \(P\) is the initial population,
- \(r\) is the rate of decline,
- \(x\) is the number of years,
- \(y\) is the population after \(x\) years.

For this specific case:
- \(P = 15,000\),
- \(r = 0.02\), so \(1 - r = 0.98\).

Putting this together, we get:

\[
y = 15,000 \cdot (0.98)^x
\]

Therefore, the correct answer is:

\[
y = 15,000 \cdot (0.98)^x
\]

So, the correct option from your list is:

**y = 15,000 ⋅ (0.98)^x**.