Questions: The rabbit population at the city park increases by 15% per year. If there are initially 288 rabbits in the city park. a) Write a model for the population (y) in terms of years ( t ). y=

Solution Steps

To model the rabbit population, we can use the formula for exponential growth: \( y = y_0 \times (1 + r)^t \), where \( y_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in years. In this case, the initial population \( y_0 \) is 288 rabbits, and the growth rate \( r \) is 15\% or 0.15. We need to express the population \( y \) as a function of time \( t \).

The rabbit population can be modeled using the exponential growth formula:

\[ y = y_0 \times (1 + r)^t \]

where:

• \( y_0 = 288 \) (initial population),
• \( r = 0.15 \) (growth rate),
• \( t \) (time in years).

To find the population after \( t = 5 \) years, we substitute the values into the model:

\[ y = 288 \times (1 + 0.15)^5 \]

Calculating \( (1 + 0.15)^5 \):

\[ (1.15)^5 \approx 2.011357 \]

Now, substituting this back into the equation:

\[ y \approx 288 \times 2.011357 \approx 579.2709 \]

Rounding the result to four significant digits gives us:

\[ y \approx 579.3 \]

Final Answer