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?

(1 point)
Responses

y=15,000⋅(2)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
𝑦
=
15
,
000
⋅
(
0.98
)
𝑥
y is equal to 15 comma 000 times 0 point 9 8 to the x th power

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

y = 15,000 ⋅ 0.98x

Answer 1

To model the population decline of a city with a declining rate of 2% per year, we can start with the initial population and apply the rate of decline.

The population after x years can be modeled by the equation:

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

where:
- \( P \) is the initial population (15,000 in this case),
- \( r \) is the rate of decline (2% or 0.02),
- \( (1 - r) \) is the remaining population after decline (which is \( 1 - 0.02 = 0.98 \)).

Substituting the values, we get:

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

Thus, the correct equation that models the population \( y \) after \( x \) years is:

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

This corresponds to the option:

**y = 15,000 ⋅ (0.98)^x**