Solve Soft Drink Equation Homework Statement

[DMOC]

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:

$$\frac{dB}{dt} = M*B (1 - \frac{B}{M})$$

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:

$$B = 500t - \frac{40*B*t}{M}$$

I came at this because:

$$t = 0, B = 0$$

$$\frac{dB}{dt} = 500$$ when $$t = 0$$

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.

 

[LCKurtz]

The unconstrained exponential growth equation is:

$$\frac{dB}{dt} = M*B (1 - \frac{B}{M})$$

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:

$$B = 500t - \frac{40*B*t}{M}$$

I don't have time right now to analyze whether your differential equation is correct in the first place or not, but one thing for sure is that your solution isn't. You have this unknown B(t) and you don't integrate it to get B(t)*t. Assuming your DE is correct, you would need to solve it for B(t), probably by separation of variables.

 

[DMOC]

When you refer to DE, you're talking about the first equation I have, right? I copied that one straight off of my notes. :)

So for my solution, I have to take this D.E. I have and solve for B. Does that mean I have to do stuff like take the integrand of both sides? (so that dB/dt becomes B(t)? ).

 

[ideasrule]

I don't think your first equation is correct. You're probably meant to derive the equation from scratch. Focus on these two clues for now:

-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

The first sentence means that dB/dt has to be 0 when M is equal to a certain value proportional to B. You should get an equation very similar to the one in your notes, but with an extra constant.

 

[LCKurtz]

When you refer to DE, you're talking about the first equation I have, right? I copied that one straight off of my notes. :)

So for my solution, I have to take this D.E. I have and solve for B. Does that mean I have to do stuff like take the integrand of both sides? (so that dB/dt becomes B(t)? ).

Does solving a DE by the method of "separation of variables" ring a bell?

 

[DMOC]

The words "sep. of variables" doesn't ring a bell but the method does.

I was eventually able to solve the full problem, ending up with:

$$B(t) = \frac{Me^{\frac{-40t}{M}} - M}{-0.08}$$

Thanks everyone. :)

 

[DMOC]

Okay guys apparently this problem has additional steps.

I know this equation is right, but now I'm asked "If the advertising budget only allows $100 per day, what price should I charge (per bottle) to start having a profit in two days?

So does this mean that I have to plug in 2 for t in the preceding equation, set it equal to zero, then find out what M is? That will give me the amount of mine that gives me a break even - no profit, no loss. Is this the right analysis?

EDIT: Okay it looks like I just have to know how much profit I get if I charge a bottle at a certain rate. I'm not sure how to do this. Derivative?