(a)

Let the number of people who have heard the rumor be , and those havenot be . Fraction of the people who have heard the rumor .

Fraction of the people who have not heard the rumor .

.

Fraction of the people who have not heard the rumor is .

The rate of spread is proportional to the product of fraction of the people who have heard the rumor and fraction of the people who have not heard the rumor.

, where is a proportional constant.

(b)

The logistic differential equation is , where is a carrying capacity.

Compare with logistic differential equation.

Here, and .

The solution logistic model equation is , where .

Substitute and in .

.

Substitute , and in .

.

The solution of differential equation is .

(c)

The town has total people .

At 8 AM the number of people heard the rumor is .

Consider for initial time 8 AM.

Fraction of the people who have heard the rumor is .

By noon the half of the people heard the rumor.

Here .

.

Substitute the initial value in .

.

Substitute in .

.

Substitute in .

.

At what time will of the population have heard the rumor.

.

.

When , the time is 8 AM.

, the time is

.

of the rumor has spread by about .

(a) , where is a proportional constant.

(b) The solution of differential equation is .

(c) of the rumor has spread by about .