Solve the problem.
-At the beginning of an advertising campaign for a new product in a city of 500,000 people, no one is aware of the product. After 10 days, 100,000 people are aware of the product. If N = N(t) is the number of people (in
Thousands) who are aware of the product t days after the beginning of the advertising campaign, solve the
Following differential equation for N(t):

Question

Quizplus · Accepted Answer

The differential equation for exponential growth is dN/dt = kN, where k is the growth rate constant. We can find k using the initial condition given: 
N(10) = 100

N(t) = N(0)e^kt

100 = N(0)e^(10k)

N(0) = 100/e^(10k)

Since the city has a population of 500,000, we can rewrite N(t) in terms of thousands:

N(t) = 500N(0) = 500(100/e^(10k)) = 50,000/e^(10k)

To find k, we differentiate N(t) with respect to t:

dN/dt = -500k(50,000/e^(10k))

Using the initial condition given, we have:

dN/dt = kN = k(100) = -500k(50,000/e^(10k))

Simplifying, we get:

k = ln(2)/20 ≈ 0.034657

Therefore, the differential equation for N(t) is:

dN/dt = 0.034657N

This matches answer choice B. Let N(t) be the number of people who are aware of the product t days after the beginning of the advertising campaign. According to the problem, at the beginning of the campaign, t=0 and N(0) = 0. After 10 days, N(10) = 100. Hence, we can write:

dN/dt = k*N

where k is a constant that we need to determine. To do that, we use the initial condition N(0) = 0:

N(0) = 0 => k*0 = 0 => k = 0

Therefore, the solution of the differential equation is:

N(t) = 0*exp(kt) = 0

Which means that no one is aware of the product at any time. This is clearly not the case, so we need to modify the differential equation. We know that after 10 days, N(10) = 100. Using this information, we can determine k:

N(10) = 100 => k*10 = ln(2) => k = ln(2)/10

Thus, the differential equation becomes:

dN/dt = (ln(2)/10)*N

The solution of this differential equation with the initial condition N(0) = 0 is:

N(t) = 100*(2^(t/10)-1)

Therefore, the correct answer is B.

Solve the Problem. -At the Beginning of an Advertising Campaign for a New