Questions: Exponential Growth Vectored Instruction Rabbit populations can increase by 45% every 10 days. If there are 12 rabbits on a farm, how many rabbits will there be on the farm after 90 days? Future Amount = 12(1+0.45)^9 Future Amount = [?] rabbits Round to the nearest whole number:

Solution Steps

To solve this problem, we need to calculate the future population of rabbits using the formula for exponential growth. The initial population is 12 rabbits, and the growth rate is 45% every 10 days. We need to find the population after 90 days, which means we will apply the growth rate 9 times (since 90 days divided by 10 days per period equals 9 periods). We will use the formula:

\[ \text{Future Amount} = \text{Initial Amount} \times (1 + \text{Growth Rate})^{\text{Number of Periods}} \]

We start with an initial rabbit population of \( P_0 = 12 \) rabbits. The growth rate is \( r = 0.45 \) (or 45%) every 10 days.

To find the number of growth periods in 90 days, we calculate: \[ n = \frac{90 \text{ days}}{10 \text{ days/period}} = 9 \text{ periods} \]

Using the formula for exponential growth: \[ P = P_0 \times (1 + r)^n \] we substitute the values: \[ P = 12 \times (1 + 0.45)^9 \] Calculating this gives: \[ P \approx 340.0112 \]

Rounding the future population to the nearest whole number, we find: \[ P \approx 340 \]

Final Answer