Thread: [SOLVED] Diffucult intro to calculus problem

Coming from Calculus 1, section on introduction to differential equations.

Corruption in Government-
The number of people implicated in a certain major government scandal increases at a rate jointly proportional to the number of people already implicated and the number involved who have yet been implicated. Suppose 7 people were implicated when a Washington newspaper first made the scandal public, that 9 more were implicated over the next 3 months, and that another 12 people were implicated during the following 3 months. Approximately how many people are involved in the scandal?
[Warning: This problem will test your algebraic ingenuity!]

Anyone have any clues on how to begin this bad boy?

Anyone have any clues on how to begin this bad boy?

do a google search on the calculus model for logistic growth ... and yes, it's a bad boy for a beginner.

Originally Posted by Sm10389

Coming from Calculus 1, section on introduction to differential equations.

Corruption in Government-
The number of people implicated in a certain major government scandal increases at a rate jointly proportional to the number of people already implicated and the number involved who have yet been implicated. Suppose 7 people were implicated when a Washington newspaper first made the scandal public, that 9 more were implicated over the next 3 months, and that another 12 people were implicated during the following 3 months. Approximately how many people are involved in the scandal?
[Warning: This problem will test your algebraic ingenuity!]

Anyone have any clues on how to begin this bad boy?

The number implicated n satisfies the differential equation:

dn/dt=k n(N-n)

where k is a constant of proportianality, and N is the tital number involved.

9 additional people implicated in the first 3 months gives a rate of increase of ~3 per month after 1.5 months and the number involved at that time is ~=7+4.5=11.5.

This gives you your first equation:

3=k 11.5(N-11.5)


Over the nest three months 12 new people were implicated so the rate of implication is now ~4 per month, at 4.5 months when the number implicated is ~=16+6=22.

This gives us out second equation:

4=k 22(N-22).

Now solve the pair of equations for k and N

CB

why is there an n outside the parenthesis?
why do you use 1.5 months and 4.5 months?
how would i solve k and n with two variables in one equation?

i appreciate your help!

Originally Posted by Sm10389

why is there an n outside the parenthesis?
why do you use 1.5 months and 4.5 months?
how would i solve k and n with two variables in one equation?

i appreciate your help!

1. There is an n outside the (N-n) term as you are told the rate of new implications is proportional to the number already implicated (n) times the number involved but not yet implicated (N-n).

2. As the rate of implications is changing the average rate of new implications is more representative of the middle of the period over which the number of new implications is quoted (the first 3 months, and the second three months).

3. You have two equations in two unknowns. If you make the variables x=k and y=Nk they become simultaneous linear equations and you should know how to solve them for x and y, and then find k and N.

CB

so then would both N and K be rather large numbers? After doing this problem, I got that N was 163,438 and K = 1.44*10^7 but those seem rather large . . .