For the first week, the spread of a rumor is proportional to the number of people who have heard the rumor. Find the particular solution to the differential equation for N, the number of people who have heard the rumor as a function of time in days, t, if 15 people have heard it at time t = 0, and 210 people have heard it at time t = 5.

Question

Quizplus · Accepted Answer

We are given that the spread of the rumor is proportional to the number of people who have heard it. This means that the rate of change of N with respect to time, dN/dt, is proportional to N.

We can write this as the differential equation:

dN/dt = kN

where k is the constant of proportionality.

To find the particular solution, we can use the initial condition that 15 people have heard the rumor at t = 0. This means that N(0) = 15.

We can also use the fact that 210 people have heard the rumor at t = 5. This means that N(5) = 210.

To solve the differential equation, we can separate variables:

dN/N = k dt

Integrating both sides:

ln(N) = kt + C

where C is the constant of integration.

To solve for C, we can use the initial condition:

ln(15) = k(0) + C

C = ln(15)

So the equation becomes:

ln(N) = kt + ln(15)

To solve for k, we can use the second initial condition:

ln(210) = k(5) + ln(15)

k = (ln(210) - ln(15))/5

k = ln(14)

So the final equation is:

ln(N) = ln(14)t + ln(15)

N = e^(ln(14)t + ln(15))

N = 15e^(ln(14)t)

N = 15(14^t)

Therefore, the correct choice is C.

For the First Week, the Spread of a Rumor Is