Question

The initial size of the population is 300. After 1 day the population has grown to 800. Estimate the population after 6 days. (Round your answer to the nearest whole number.)

Answer 1

Solution :

Given initial population = 300

Final population after 1 day = 800

Number of days = 6

∴ $\frac{dP}{dt} =kt^{1/2}$

P(0) = 300    P(1) = 300

We need to find P(8).

$dP = kt^{1/2} dt$

$\int 1 dP = \int kt^{1/2} dt$

$P(t) = k \left(\frac{t^{3/2}}{3/2}\right)+c$

$P(t)= \frac{2k}{3}t^{3/2} + c$

When P(0) = 300

$300 = \frac{2k}{3} (0)^{3/2} + c$

∴ c = 300

∴ $P(t)= \frac{2k}{3}t^{3/2} + 300$

When P(1) = 800

$800 = \frac{2k}{3} (1)^{3/2} + 300$

$500 = \frac{2k}{3}$

∴ k = 750

$P(t)= 500t^{3/2} + 300$

So, P(8) is

$P(t)= 500(8)^{3/2} + 300$

        = 11,614

So the population becomes 11,614 after 8 days.