G Suppose that when a certain lake is stocked with fish, the birth and death rates ˇ and ı are both inversely propor- (a) Show that 1 p2 P. t/D 2ktC P0 ; where k is a constant. (b) If P0 D 100 and after 6 months there are 169 fish in the lake, how many will there be after 1 year?

k=1

P(12)=256

Step-by-step explanation:

The model for population involving birth and death is given as:

where b=birth rate and d=death rate.

If the birth and death rate are inversely proportional to

where A and B are constants of variation.

Substituting b and d into our model

where A-B=k, another constant

Simplifying using indices

Next, we Separate Variables and Integrate both sides

where C is the constant of integration

When t=0, P(t)=, C=

as required.

(b)If =100, t=6 months, P(t)=169

Since we have found the constant k, we can then calculate the population after 1 year. Note that we use 12 months since we used month earlier to get k.

Therefore the population after a year is 256.

$15 each

step-by-step explanation:

$45/3 children= $15 each