Math Forum / Mathematics / Algebra Help / September 2008

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This looks like a linear programming problem to me.

All numbers are in thousands.

Let t be the number of TV spots. So the total cost ( in thousands ) for
TV is 5t and the number of cans sold due to TV is 300t.

Let m be the number of mag ads. So the total mag ad cost is 2m and the
sales due to mag ads is 500m.

The profit is P(t, m) = 15t + 25m.
(That is, 0.05*300,000*t + 0.05*500,000*m.)

We have five constraints:

t >= 0;
t <= 70;
m >= 0;
m <= 50;
5t + 2m <= 100.

If you graph the points that satisfy all of these constraints ("t"
being the "x - axis"), you'll get a triangle with vertices (0, 0),
(20, 0) and (0, 50).

P is maximized at one of these vertices so just see which one gives the
largest value for P - I got (0, 50).

To see that this is correct, consider the general line c = 15t + 25m
and convince yourself that c is biggest at the highest point in the
triangle touched by the line - i.e. (0, 50).

This defines two relationships.

This defines a third.

This is the expression you want to maximize.  But you need the
following constraints also.

    5000(x1) <= 70,000
    2000(x2) <= 50,000

The .05 profit tells you that for every 5,000 spent on TV ads the
profit increases 15,000 for a net of 10,000.  For every 2,000 spent on
magazines ads the profit increases 25,000 for a net of 23,000.
However, you don't really need this.

Are you sure you copied the problem correctly?  Magazine ads are
cheaper and have greater impact so obviously you spend the max on them
(50,000) and the remaining 50,000 on TV ads.  So x1 is 14 and x2 is
25.