A flock of sheeps counts 1000. The flock grows by 25% each year. Every fall 10% gets send to the slaughterhouse. How big will the flock be in 10 years?

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- Feb 19th 2011, 03:10 PM #1

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- Feb 19th 2011, 03:20 PM #2
I'm not sure how your growth and loss will be counted. I am assuming the growth occurs before the slaughter.

Let the initial amount of sheep be $\displaystyle P_0$

After the growth there will be $\displaystyle P_0(1+0.25) = 1.25P_0$

It is important to notice that the 10% slaughtered is based on the new amount: $\displaystyle P_1 = 0.9 \cdot (1.25P_0)$

This is going to be the initial value in the second year (the values of t are for the end of the year)

$\displaystyle P_2 = 0.9(1.25P_1) = 0.9(1.25 \cdot 0.9(1.25P_0) = 0.9^2 \cdot 1.25^2 \cdot P_0$

$\displaystyle P_3 = 0.9(1.25P_2) = 0.9^3 \cdot 1.25^3 \cdot P_0$

There is a pattern emerging here which is

$\displaystyle P_n = 0.9^n \cdot 1.25^n \cdot P_0$

To answer your question sub $\displaystyle n=10$ and $\displaystyle P_0 = 1000$