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(0.91)t
y is equal to 78 times 0 point 9 1 to the t th power

y=78(1.09)t
y is equal to 78 times 1 point 0 9 to the t th power

y=78(0.09)t
y is equal to 78 times 0 point 0 9 to the t th power

y=78(0.9)t
y is equal to 78 times 0 point 9 to the t th power

Answer 1

To create an exponential decay function for the population of hyenas, we need to account for the rate of decline, which is 9% per year.

This means that 91% of the population remains each year. Therefore, we can express the population of hyenas after \( t \) years using the formula:

\[ y = y_0 \cdot (1 - r)^t \]

where:

• \( y \) is the population at time \( t \),
• \( y_0 \) is the initial population (78 hyenas),
• \( r \) is the rate of decline (0.09 in this case),
• \( t \) is the time in years.

Substituting the values in, we have:

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

Thus, the exponential decay function is:

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

The correct response is:

y = 78(0.91)^t
(y is equal to 78 times 0.91 to the t-th power)