Question

In a certain geographic location, a herd of elephants is declining at a rate of four percent every year. If there are currently 62 elephants in the herd, create an exponential decay function to model the problem. Let Y represent the number of elephants after t years.
Options:
A. y = 62(1.04)t
B. y = 62(0.6)t
C. y = 62(0.96)t
D. y = 62(0.04)t

Answer 1

To model the decline of the herd of elephants, we use the exponential decay function. The general form for exponential decay can be expressed as:

\[ Y = Y_0 \cdot (1 - r)^t \]

where:
- \( Y_0 \) is the initial quantity (which is 62 elephants in this case),
- \( r \) is the rate of decay (4% or 0.04),
- \( t \) is time in years, and
- \( Y \) is the quantity after \( t \) years.

Given that the herd is declining at a rate of 4% per year, we can substitute the values in:

- Initial quantity (\( Y_0 \)): 62
- Rate of decay (\( r \)): 0.04

We can calculate \( 1 - r \):

\[ 1 - r = 1 - 0.04 = 0.96 \]

Now, substituting these values into the equation gives us:

\[ Y = 62 \cdot (0.96)^t \]

This matches with option C.

So, the exponential decay function to model the decline of the elephants is:

**C. \( y = 62(0.96)^t \)**.