A group of hyenas is called a cackle. In a certain geographic location, a cackle of hyenas is declining at a rate of nine percent every year. If there are currently 78 hyenas in the cackle, create an exponential decay function to model the problem.(1 point)

Question

Responses

y=78(0.9)t
𝑦
=
78
(
0.9
)
𝑡
y equals 78 left parenthesis 0.9 right parenthesis superscript t baseline

y=78(0.91)t
𝑦
=
78
(
0.91
)
𝑡
y equals 78 left parenthesis 0.91 right parenthesis superscript t baseline

y=78(1.09)t
𝑦
=
78
(
1.09
)
𝑡
y equals 78 left parenthesis 1.09 right parenthesis superscript t baseline

y=78(0.09)t

Answer 1

To model the population of the hyenas using an exponential decay function, we start with the initial number of hyenas and account for the annual decline rate.

Given:

• Initial population, \( P_0 = 78 \)
• Decline rate = 9% (which means the population retains 91% of its previous count each year)

The exponential decay function is generally given by:

\[ y = P_0 (1 - r)^t \]

where:

• \( y \) is the population after time \( t \),
• \( P_0 \) is the initial population,
• \( r \) is the decay rate (as a decimal), and
• \( t \) is time in years.

In this case, the decay rate \( r = 0.09 \), and thus the function becomes:

\[ y = 78(1 - 0.09)^t \] \[ y = 78(0.91)^t \]

So, the correct exponential decay function for the scenario described is:

\[ y = 78(0.91)^t \]

Among the provided choices, the correct answer is:

y = 78(0.91)^t.