stuck on this problem

Question

The rate r at which a disease spreads in a population of size P is jointly proportional to the number x of infected people and the number P − x

 who are not infected. An infection erupts in a small town with population P = 6,000.

(a)

Write an equation that expresses r as a function of x. (Use k for the constant of proportionality and P for the population.)

 

(b)

Compare the rate of spread of this infection when 40 people are infected to the rate of spread when 1,300 people are infected. Which rate is larger? By what factor? (Round your answer to one decimal place.)

The infection rate when 1,300 people are infected is about times  ---Select--- smaller than larger than when 40 people are infected.

(c)

Calculate the rate of spread when the entire population is infected.

r = 

Why does this answer make intuitive sense?

Answer 1

a)

r = kx(P-x)

r = kx(6000-x)

r = 6000kx - kx^2

b)

x=40 or 1300

r = 6000k(40) - k(40)^2 = 240,000k - 1600k = 238,400k for x=40

r = 6000k(1300) -k (1300)^2 = 7800,000k - 1690,000k= 6,110,000k for x=1300

r is greater for x=1300 than for x=40

6,110,000/238,400 = 61100/2384 = about factor of 25.6

c) x=6000

r = 6000k(6000) -k(6000)^2 = 0

the rate of spread reduces to zero, because there is no one left to infect