Question
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 each year. If there are currently 78 hyenas in the cackle, create an exponential decay function to model the problem.
A. y=78(0.9)^t B.78(0.91)^t C.78(0.09)^t D. 78(1.09)^t
A. y=78(0.9)^t B.78(0.91)^t C.78(0.09)^t D. 78(1.09)^t
Answers
GPT-4o mini
To model the exponential decay of the cackle of hyenas, we can use the formula:
\[ y = a(1 - r)^t \]
where:
- \( y \) is the amount remaining after \( t \) years,
- \( a \) is the initial amount (in this case, 78),
- \( r \) is the decay rate (as a decimal), and
- \( t \) is the time in years.
Given that the decay rate is 9%, we convert that percentage to a decimal:
\[ r = 0.09 \]
Thus, the equation becomes:
\[ y = 78(1 - 0.09)^t \]
\[ y = 78(0.91)^t \]
Looking at the choices provided:
- A. \( y=78(0.9)^t \)
- B. \( 78(0.91)^t \)
- C. \( 78(0.09)^t \)
- D. \( 78(1.09)^t \)
The correct choice is **B. \( 78(0.91)^t \)**.
\[ y = a(1 - r)^t \]
where:
- \( y \) is the amount remaining after \( t \) years,
- \( a \) is the initial amount (in this case, 78),
- \( r \) is the decay rate (as a decimal), and
- \( t \) is the time in years.
Given that the decay rate is 9%, we convert that percentage to a decimal:
\[ r = 0.09 \]
Thus, the equation becomes:
\[ y = 78(1 - 0.09)^t \]
\[ y = 78(0.91)^t \]
Looking at the choices provided:
- A. \( y=78(0.9)^t \)
- B. \( 78(0.91)^t \)
- C. \( 78(0.09)^t \)
- D. \( 78(1.09)^t \)
The correct choice is **B. \( 78(0.91)^t \)**.