CPM Homework Help : APCALC Problem 11-99

The population $P$ of mountain lions grows at a rate of $\frac { d P } { d t }=\frac { 1 } { 160 }P(40 − P)$ lions per year. At time $t = 0$ , there are $12$ lions. What is the population after three years?

Step 1:

$\frac{160dP}{P(40-P)}=dt$

Step 2:

Partial fraction decomposition needs to be used.

$\frac{A}{P}+\frac{B}{40-P}=\frac{1}{P(40-P)}$

Step 3:

$160\int\Big(\frac{A}{P}+\frac{B}{40-P}\Big)dP=\int dt$

Step 4:

$160(A\ln|P|-B\ln|40-P|)=t+C$

Step 5:

Solve for $C$ when $t = 0$ and $P = 12$ .

Step 6:

Substitute your value for $C$ into the equation from Step 4 along with $t = 3$ . Then solve for $P$ .