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## Homework Statement

A soft-drink manufacturing company introduces a new product. The company's salespeople want to predict the number of bottles per day they will sell as a function of the number of days since the product was introduced. One of the parameters will be the amount per day spent on advertising. Here are some assumptions the salespeople make about the sales.

-The dependant variable is B bottles per day; the independant variable is t days.

-They will spend a fixed amount, M dollars per day, on advertising.

-Part of M, an amount porportional to B, maintains present sales

-The rate of change of B, dB/dt, is directly proportional to the rest of M.

-Advertising costs need to be $80 per day to maintain sales of 1,000 bottles per day.

-Due to advance publicity, dB/dt will be 500 bottles per day when t=0, independant of M.

Find an equation for B as a function of t. Then show the effect of spending various amounts, M, on advertising.

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## Homework Equations

The unconstrained exponential growth equation is:

[tex]\frac{dB}{dt} = M*B (1 - \frac{B}{M})[/tex]

where B is population (or in this case, I guess it's bottles per day?) and M is the maximum sustainable population (or bottles?).

## The Attempt at a Solution

My attempt at the equation is something like this:

[tex]B = 500t - \frac{40*B*t}{M}[/tex]

I came at this because:

[tex]t = 0, B = 0[/tex]

[tex]\frac{dB}{dt} = 500[/tex] when [tex]t = 0[/tex]

However, I am unsure if this is the correct equation based on the 6 parts given above. I'm supposed to express B in terms of M and t.