In 1859, 24 rabbits were released into the wild in Australia, where they had no natural predators. Their population grew exponentially, doubling every 6 months.

a. Determine P(t), the function that gives the population at time t, and the differential equation describing the population growth.
b. After how many years, rounded to one digit after the decimal point, did the rabbit population reach 1,000,000?
c. Determine the rate of population change, in rabbits/year, midway through the third year. (Warning: t is not 3.5, just like the year midway through the 21st century is not 2150.) Round the final answer to 2 digits after the decimal point.

a) P' = P

    where t is step of 6 months

b) 7.7 years

c)1064.67 rabbits/year

Step-by-step explanation:

The differential equation describing the population growth is

Where t is the range of 6 months, or half of a year.

P(t) would have the form of

where is the initial population

After 6 month (t = 1), the population is doubled to 48

Therefore

where t is step of 6 months

b. We can solve for t to get how long it takes to get to a population of 1,000,000:

So it would take 15.35 * 0.5 = 7.7 years to reach 1000000

c.

We need to resolve for k if t is in the range of 1 year. In half of a year (t = 0.5), the population is 48

Therefore,

At the mid of the 3rd year, where t = 2.5, we can calculate P'

rabbits/year

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

b

step-by-step explanation: