For your question, I will assume that the exposure number listed is
the number of people exposed to the Cedar Point advertisement for
everyone one unit of the advertising type purchased, and that this
rate remains constant regardless of the number of units purchased.
The first thing we need to do is analyze all types of advertising to
determine what the average cost is to expose one person to Cedar Point
advertising. To do this, we divide the cost of a unit of advertising
by the exposure that each unit receives. In the order written above
(answers rounded to the nearest cent):
Newspaper: 1500 / 80 = $18.75 per exposure
Television: 2200 / 120 = $18.33 per exposure
Radio: 750 / 45 = $16.67 per exposure
Shows: 150 / 10 = $15.00 per exposure
From this, we see that we get the best value from Shows, followed by
Radio, Television, and finally Newspaper advertising. Because Shows
is always the best value, common sense seems to dictate that we should
purchase as many units of Shows advertising as possible.
Unfortunately, there are only three units available, so this is as
much as we can possibly purchase.
Three units of Show advertising valued at $150 each costs $450 total,
and gains Cedar Point 30 (10*3) exposures. We now have $249,550
($250,000 - $450) remaining in our budget.
The next best value comes from Radio, so we buy as much of that as possible.
50 units of Radio advertising valued at $750 each costs $37,500 total,
and gains Cedar Point 2,250 (50*45) exposures, bringing the total of
exposures to 2,280 (2,250 + 30). We now have $212,050 ($249550 -
$37500) remaining in our budget.
The next best values comes from Television, so we buy as much of that as possible.
50 units of Television advertising valued at $2200 each costs $110,000
total, and gains Cedar Point 6,000 (50*120) exposures, bringing the
total of exposures to 8,280 (6,000 + 2,250 + 30). We now have
$102,050 ($212050 - $110000) remaining in our budget.
The only advertising medium remaining is Newspaper, so we spend as
much of what's remaining in our budget on this. Dividing our total
remaining budget of $102,050 by the $1,500 per unit cost of Newspaper
advertising shows that we can purchase 68 Newspaper advertising units,
to gain Cedar Point an additional 5,440 (68*80) exposures, bringing
the total number of exposures to 13,720 (5,440 + 6,000 + 2,250 + 30).
If you multiply the $1,500 per unit cost of Newspaper advertising by
the 68 units we purchased, you'll see that this total comes out to
$102,000, meaning that we still have $50 left over. Is it possible to
use this $50 somehow to get just a little bit more exposure for Cedar
Point? We've already purchased all of the other mediums available to
us, but if we freed up $1,450 worth of advertising from those other
mediums, we could purchase one more Newspaper advertisement and use
all of our alloted budget, and possibly pick up a few extra exposures.
Since Television advertising costs more than $1,450, the only way we
could raise the money is by buying less Radio and Shows advertising.
However, there is no way to factor 750 and 150 to get 1450, so it is
not possible to free up exactly $1,450. So the next step is to see if
it is possible to free up $2,950 from other advertising, so that we
can purchase two additional units of Newspaper advertising to use our
With $2,950 required, we can now free up $2,200 by purchasing one less
Television advertisement, leaving us with $750 to raise. Since the
sum of our Shows advertising is only $450, the only way we can do this
is to purchase one less Radio unit as well. So by giving up one Radio
and one Television advertisement, we can use all our money to purchase
two additional Newspaper advertisements. But is it worth while to do
this? One Television advertisement brings 120 exposures, while one
Radio advertisement yields 45, for 165 exposures lost. In exchange,
we get 80 exposures per Newspaper advertisement, for 160 exposures
total, five less than we had before. Even though we found a way to
use our last $50, this can only be done by reallocating funds to less
effective advertising, and does not provide optimal results.
Therefore, our original calculations were correct, and the optimal
allocation of advertising funds is as follows:
Shows: 3 (30 exposures
Radio: 50 (2,250 exposures)
Newspaper: 50 (6,000 exposures)
Television: 68 (5,440 exposures)
Total Exposures: 13,720
Total Budget Used: $249,950
Average Cost Per Exposure (rounded to nearest cent): $18.22
Source Referenced: Google Calculator