Assume that the growth rate is the same as in Part A. Use 2 as the initial amount. Substitute 80 000 for y and solve for t.
E
Is it reasonable to assume that the growth rate is always the same? Are there still no natural enemies for rabbits in Australia?
A
See solution.
B
y=80 000* (sqrt(30))^t
C
394 360 241
D
1859, see solution.
E
See solution.
Practice makes perfect
Let's first try to think why a sine function would not be better for this situation. Let's see what a sine function graph looks like.
We can see that the sine functions increases and decreases in different points. However, we are told that the population of rabbits had no natural enemies. Populations which do not have natural enemies do not decrease because there are no dangers for them in the wild. A sine function would not be good for modeling the growth of this population. Let's consider a linear function!
Linear growth means constant growth in equal time intervals. This would not be a good model for population of rabbits because the current number of rabbits influences the number of newborn rabbits. For example, a population of 10 rabbits will give birth to fewer rabbits than a population of 100. This is why an exponential model is better, because the more rabbits there are currently, the faster the growth.
Let's recall the formula for the exponential growth.
y=ab^t
In this form, a is the initial amount, the base b is the growth factor, and t represents time in years. We know that there were 80 thousand rabbits in 1866, which we can mark as the starting t=0.
80000=ab^0
Remember that b^0=1. With this in mind, we know the value of a!
80000=a
We also know that after 2 years, the population grew to be 2 400 000.
2 400 000=ab^2=80000b^2
With this in mind, we can find the value of b. Let's do it!
Finally, we can write the equation for the population of rabbits after t years from 1866.
y=80 000* (sqrt(30))^t
In order to make a prediction for the number of rabbits in 1871, we need to substitute 5 for t in the equation we found in Part A and calculate y. This is because 1871 is exactly 5 years after 1866. Let's do it!
This means that our best prediction is that there were 394 360 241 rabbits in the year 1871.
In Part A, we assumed the year 1866 to be the first year we started calculating from. This time, let us assume that the growth factor for rabbits is the same and equals b=sqrt(30). Our initial amount will be 2, because we want to find the year when the first pair of rabbits appeared. Let's write the equation for the population of rabbits t years after the first pair.
y=2* (sqrt(30))^t
We are told that there were 80 thousand rabbits in the year 1866 but we do not know which year after the first pair it was. We need to substitute 80 000 for y and calculate the number of years passed, y. Let's do it!
This suggests that 1866 came a little over 6 years after the first pair or rabbits. So it would mean the first pair was introduced in the year 1859. However, this model is not entirely reasonable, because we assume that the growth rate is always the same.
b=sqrt(30)
First of all, if only one pair had been introduced at the beginning, there would have been a lot of inbreeding in the first few generations of new rabbits. This makes the population prone to genetic mutations and affects the growth rate. Usually, the growth rate does not shift as much from generation to generation but there are many factors that could potentially change it.
As we stated in Part D, the growth rate can potentially change in time. This is especially true for bigger periods of time, so estimating the number of rabbits today using the same model would not be accurate as there is a difference of over 150 years in time, which is a long time considering the average lifespan of a rabbit.
b ≠ sqrt(30)
What is more, we know that the exponential growth was good for estimating the population in the early years because the rabbits had no natural enemies. However, this can change over the years and have likely changed to this day. There are many predators which threaten the existence of rabbits and this influences the population numbers by a lot.
