There is 90 black nosed rabbits in a forest in 2000. In 2003, they there is 180 black nosed rabbits in the same range. If the population of rabbits follows the exponential law, how many rabbits will be in the range 5 year from 2000?

Question

anonymous · Answer

$$P _{n}= P _{0}(1+ r)^n  (n = 1, 2, 3...)$$

anonymous · Answer

Take n=0 at 2000. Then n= 3 in 2003, so $$P _{0}= 90  and  P _{3}= 180$$
According to the exponential model (equation above) :
$$P _{3}= (1 + r)^3 P _{0}$$ that is, 180 = (1+r)^3 x 90

anonymous · Answer

it remains to solve this equation to find the proportionate growth rate, r. 
1 + r = $$\sqrt[3]{180/90}$$

anonymous · Answer

Now if you remember that the cubed root is (x)^ 1/3 then we have:
$$1+r = (180/90#)^1/3$$
or r = (180/90)^1/3 -1

anonymous · Answer

$$r= [(2)^{1/3}] -1$$

anonymous · Answer

So now you have $$P _{0}$$, r = growth rate, and number of years n= 5, so away you go!

anonymous · Answer

remember when working out on calculator to include brackets around 1/3 
r = (1.25992105) -1= 0.25992105

anonymous · Answer

to two significant figures this is 0.26

anonymous · Answer

Example: $$P _{3} - P _{0} (1 + 0.26)^3$$
$$P _{3}= 90 x (1+0.26)^3$$
$$P _{3}= 90 x (1.26)^3$$
$$P _{3}= 90 x 2.000376$$
$$P _{3}= 180.03384$$
which is a pretty good approximation

anonymous · Answer

the inaccuracies is due to the round-off in the 0.26

anonymous · Answer

P.S. If you are studying this, you must be doing well in maths! best of luck in your studies!