Question

(1 point) The time rate of change of a rabbit population P is proportional to the square root of P. At time t=0 (months) the population numbers 100 rabbits and is increasing at the rate of 10 rabbits per month. Let P′=kP12 describe the growth of the rabbit population, where k is a positive constant to be found. Find the formulas for k and for the rabbit population P(t) after t months.

Answer 1

Answer:

2√P = t + 20

Step-by-step explanation:

Rate of change of a rabbit population ∝ √P

$\frac{dP}{dt}=k\sqrt{P}$

$k=\frac{1}{\sqrt{P}}.\frac{dP}{dt}$ -----(1)

At time 't' = 0 the population of the rabbits (P) = 100 and $\frac{dP}{dt}=10$ rabbits per month.

We plug in the values in the equation (1),

$k=\frac{1}{\sqrt{100}}\times 10$

k = 1

Equation (1) becomes, $\frac{dP}{dt}=1\times \sqrt{P}$

$\frac{dP}{\sqrt{P}}=dt$

By the integration of this equation,

$\int \frac{dP}{\sqrt{P}}=\int dt$

$\frac{\sqrt{P}}{\frac{1}{2}}=t+C$

$2\sqrt{P}=t+C$

Again for t = 0 and P = 100,

$2\sqrt{100}=0+C$

C = 20

Now the integrated equation will be,

$2\sqrt{P}=t+20$

Therefore, formula representing population P after time t is,

2√P = t + 20