The time rate of change of a rabbit population P is proportional tothe square root of P. At time t = 0 (months) the populationnumbers 100 rabbits and is increasing at the rate of 20 rabbits permonth. How many rabbits will be there one year later?

121

Step-by-step explanation:

dP/dt is propotional to P^½

dP/dt = k (P^½)

When P=100, dP/dt=+20

20 = k (100^½)

20 = 10k

k = 2

dP/dt = 2 (P^½)

Reciprocate both sides:

dt/dP = 1/[2 (P^½) ]

dt/dP = ½[P^ (-½) ]

Integrate wrt P.

t = ½ [ (P^½) : ½] + c

t = P^½ + c

Using t = 0, P = 100 ... find c

0 = 100^½ + c

0 = 10 + c

c = - 10

t = P^½ - 10

Find P when t=1

1 = P^½ - 10

P^½ = 11

P = 121