Question 1174706: A company has budgeted a maximum of $600,000 for advertising a certain product nationally. Each minute of television
time costs $60,000 and each one-page newspaper ad costs
$15,000. Each television ad is expected to be viewed by 15
million viewers, and each newspaper ad is expected to be
seen by 3 million readers. The company’s market research
department advises the company to use at most 90% of the
advertising budget on television ads. How should the
advertising budget be allocated to maximize the total audience?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x = the number of one minute television ads.
y = the number of one page newspaper ads.
your objective function that you want to maximize is:
number of viewers = 15,000,000 * x + 3,000,000 * y
your constraint inequalities are:
60,000 * x + 15,000 * y <= 600,000
60,000 * x <= .9 * 600,000
x >= 0
y >= 0
using the desmos.com calculator, you will graph the opposite of these inequalities.
the area on the graph that is not shaded is your feasible region.
you will evaluate your objective function at each of corner points of the feasible region.
the corner point with the maximum number of viewers is your solution.
the graph looks like this:
your corner points are at (0,40), (9,4), (9,0).
your objective function is 15,000,000 * x + 3,000,000 * y
at the point (0,40), that becomes 15,000,000 * 0 + 3,000,000 * 40 = 120,000,000.
at the point (9,4), that becomes 15,000,000 * 9 + 3,000,000 * 4 = 147,000,000.
at the point (9,0), that becomes 15,000,000 * 9 + 3,000,000 * 0 = 135,000,000.
your maximum number of viewers = 147,000,000 at the point (9,4)
that's 9 one minute tv ads plus 4 one page newspaper ads.
your constraints must all be met at the point (9,4)
60,000 * x + 15,000 * y = 60,000 * 9 + 15,000 * 4 = 600,000 which is <= 600,000 so this constraint is met.
x = 9 and y = 4, so x >= 0 and y >= 0 these constraints are met.
.9 * 600,000 = 540,000
60,000 * x = 60,000 * 9 = 540,000 which is smaller than or equal to 540,000 so this constraint is true.
all constraints are met at the corner point of (9,4) where the number of viewers is equal to 147,000,000.
your solution is that the advertising budget should be allocated to 9 one minute tv ads and 4 one page newspaper ads to maximize the number of viewers at 147,000,000.
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