Yolanda owns 4 rabbits. She expects the number of rabbits to double every year. b) write and equation to model this situation?
The Correct Answer and Explanation is:
Equation:
To model the growth of the rabbit population, we use the formula for exponential growth:P(t)=P0⋅2tP(t) = P_0 \cdot 2^tP(t)=P0⋅2t
Where:
- P(t)P(t)P(t) is the number of rabbits after ttt years,
- P0P_0P0 is the initial number of rabbits,
- ttt is the number of years,
- The base 2 represents the fact that the population doubles each year.
Since Yolanda starts with 4 rabbits, P0=4P_0 = 4P0=4.
So, the equation becomes:P(t)=4⋅2tP(t) = 4 \cdot 2^tP(t)=4⋅2t
Explanation
The situation described involves Yolanda starting with 4 rabbits and expecting the number of rabbits to double every year. This is a classic case of exponential growth, where the rate of increase is proportional to the current amount—in this case, the number of rabbits.
To write an equation modeling this situation, we start with the general exponential growth formula:P(t)=P0⋅rtP(t) = P_0 \cdot r^tP(t)=P0⋅rt
Here, P(t)P(t)P(t) is the population after ttt years, P0P_0P0 is the starting number, and rrr is the growth rate. Since the number of rabbits doubles each year, the growth rate rrr is 2.
Yolanda starts with 4 rabbits, so P0=4P_0 = 4P0=4. Substituting into the formula, we get:P(t)=4⋅2tP(t) = 4 \cdot 2^tP(t)=4⋅2t
This equation tells us how many rabbits Yolanda will have after any given number of years. For example:
- After 1 year: P(1)=4⋅21=8P(1) = 4 \cdot 2^1 = 8P(1)=4⋅21=8
- After 2 years: P(2)=4⋅22=16P(2) = 4 \cdot 2^2 = 16P(2)=4⋅22=16
- After 3 years: P(3)=4⋅23=32P(3) = 4 \cdot 2^3 = 32P(3)=4⋅23=32, and so on.
The population keeps increasing faster over time because each year’s growth builds upon the previous total. This type of growth is very different from linear growth, where the same number is added each year. Instead, exponential growth accelerates—doubling the population every year means it grows very rapidly.
This model assumes ideal conditions: no rabbits die, and there are enough resources to support unlimited growth. While not realistic over long periods, this equation is a good mathematical model for understanding the concept of doubling and exponential increase.
