A population of 20 rabbits is introduced to a small island. The population increase at a rate of 60% per year.
(A) Find a function of F(t) that represents the number of rabbits on the island after t years.
(B) Suppose 100 rabbits were introduced to the island and G(t) is the number of rabbits after t years. Describe the stretching or shrinking operations that must be performed on the function y= F(t) to obtain the function
y=G(t). Write the new function y= G(t) that results from this transformation.
F(t) = 20 * 1.6^t
G(t) = 100 * 1.6^t = 5 F(t)
once five times as many, always five times as many :)
so how would you describe the stretching?
(A) To find a function that represents the number of rabbits on the island after t years, we can start with the initial population of 20 rabbits and then apply the growth rate of 60% per year.
Let's call the function F(t) the number of rabbits after t years. Initially, F(0) = 20, and after one year, the population will increase by 60%, which means it will be 1.6 times the previous year's population. So, after one year, the number of rabbits will be F(1) = 20 * 1.6.
Generalizing this, we can write the function as:
F(t) = F(t-1) * 1.6
Now, let's simplify this function further:
F(t) = F(t-1) * 1.6
= (F(t-2) * 1.6) * 1.6
= F(t-2) * 1.6^2
Continuing this pattern, we get:
F(t) = F(t-k) * 1.6^k
We need to find the value of k to represent the population after t years. Since the population increases 60% per year, the value of k will be t, so:
F(t) = F(t-t) * 1.6^t
= F(0) * 1.6^t
= 20 * 1.6^t
Therefore, the function that represents the number of rabbits on the island after t years is:
F(t) = 20 * 1.6^t
(B) To obtain the function G(t) that represents the number of rabbits after t years when 100 rabbits are initially introduced, we need to perform stretching or shrinking operations on F(t) = 20 * 1.6^t.
Since there are initially 100 rabbits instead of 20, we need to multiply the function by a scaling factor. The scaling factor can be found by dividing the initial population of G(t) (100) by the initial population of F(t) (20). So the scaling factor is 100 / 20 = 5.
Therefore, the new function G(t) that results from this transformation is:
G(t) = 5 * F(t)
= 5 * (20 * 1.6^t)
= 100 * 1.6^t