Question

1. A seaside village had a stable population of 1,000 families for several decades until a new species of seaweed invaded the coast. The aroma emanating from the seaweed is pungent and obnoxious. The villagers tried several chemical and biological treatments to eliminate the invasive seaweed, but nothing has worked. The aroma is overwhelming. After a couple of years of frustration, the villagers started moving out of the village. Each year the fraction of families moving out of the village has been steady at 5% of the families still inhabiting the odoriferous seaside village. Use No as the original number of families living in the village (1,000); N as the number of families still living in the village; k as the fraction of families leaving the village every year (0.05 yr-)); t as the time in years when the emigration started. Derive a relation for the population of families (N) still inhabiting the village as a function of the time (t) in years. a. First, write in symbols a rate of families balance (write symbols in appropriate boxes): Rate of families IN per year Rate of families OUT per year Rate of ACCUMULATION of families per year b. Write the rate balance as a differential equation. || c. Write the appropriate boundary condition(s). d. Solve to write the fraction of original families (N/N.) still inhabiting the village after t years. e. Write the characteristic time (T) for the emigration; i.e. the time for the population of families to decrease to 37% (1/e) of its original value. f. Sketch and label fraction of original families as f(t). g. Illustrate one, two, and three characteristic times.

Answer 1

A seaside village has a stable population of 1000 - No=1000 Because of obnoxious ordour, each year, the paction of famelees moving out of the Village has been steady at 5% al the famplies still inhabitating. - kFoo = 0.05 Rate of families In per year = IN IN Rate of families OUT per year = NOUT Rate of families accumulated per year - d aca este in at - at © Here, at the starting of emigration - protially, Noodls families= 1000 (NO) = 1000) This is the boundary condition

- (ol) Here, IN =kN negative symbol is because ! de ma co 8 d= rkN and 1/ den 6008) N dw = -0.05 dt Integrate on both the sedes ~ Sen S 60.05) at a) en (NJ) = -0.05 t +K [ats en laten » Nae oostek constant "c" =) Nee-oostek Het let ek be some other -0.0SET Natace

We have the boundary (o)=(COOH condition => 1000 = ce-0.005() -> c=1000 Actual solection is st N(E)= 1000e 1(e) we need characteristic time (1) I time for the population of famples to decrease to 37% (2) of its orginal Value oost lear "E= N or = No e Forest > -0.ostl => etcerooster =) pa dos = 20 N= 20 yeary] 1 (9) 2n = 40 years & 39=60 years
Now, we shall see the graph:

the zoomed version of above graph is

-80- -80 40 60

<o8642 -14 -12 -10 -8 -6 -4 - 2 0 8 10 12 14 16 18-