To model the rabbit population growth, we can use an exponential function. The general formula for exponential growth is given by:
\[ P(t) = P_0 \cdot r^t \]
where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth factor (how many times the population grows each period),
- \( t \) is the time in months.
In this case:
- The initial population \( P_0 = 15 \) (the starting number of rabbits),
- The population doubles every month, so the growth factor \( r = 2 \) (the population multiplies by 2 each month),
- We want to find \( P(t) \) for \( t = 7 \) months.
Substituting these values into the formula, we get:
\[ P(t) = 15 \cdot 2^t \]
Now, we can determine the number of rabbits after 7 months by substituting \( t = 7 \) into the equation:
\[ P(7) = 15 \cdot 2^7 \]
Calculating \( 2^7 \):
\[ 2^7 = 128 \]
Now plug this value back into the equation:
\[ P(7) = 15 \cdot 128 \] \[ P(7) = 1920 \]
Therefore, after 7 months, the rabbit population will be \( 1920 \) rabbits.