A newspaper ad cost s $300 and reaches 6000 readers. A 30 second radio commercia

Question

A newspaper ad costs $300 and reaches 6000 readers.

A 30 second radio commercial costs $800 and is heard by 10,000 listeners.

A 15 second TVcommercial costs $1,2 00 and is seen by 11,000 viewers

Last month,

to advertise the opening of his new

pizzeria,

Luigi

bought some

newspaper

ads,

some radio commercials and som

e TV commercials. He bought

twice as many radio

commercials as n

ewspaper ads. He spent $11,900 and his message reached 152,000 people

(ignoring situations where the

same person saw more than one ad)

. How many newspaper ads,

radio commercials and TV commercials did Luigi buy?

2.
In 5 weeks,
Luigi will open a sandwich shop.
H
e
has an $18,000 advertising budget.
At most, he
can purchase 5 newspaper ads. If he limits himself to no more than 20 radio and TV
commercial
s combined, how many newspaper ads, radio commercials and TV commercials
should Luigi buy to max
imize the number of people who heard about his sandwich shop?
CAUTION, you can cannot buy a fractional part of an ad!

Explanation / Answer

1..

Let x be the number of newspaper ads

y => the number of radio commercials

z => the number of tv commercials

Since he bought twice as many radio commercials as newspaper ads, we have

y=2x ------------- naming this equation as A.

Now, with the cost information we formulate the equation,

300 x + 800 y + 1200 z = 11,900 ------------------------- B

i.e He spent $11,900 in total for various ads.

Ads reached as many as 1,52,000 people. So we have,

6000 x + 10000 y + 11000 z = 152000 ---------------------- C

Thus we have a set of equations, Now substitute equation A in equation B. so we get,

300 x + 800 * 2x + 1200 z = 11,900

1900 x + 1200 z = 11,900 ----------------------D

dividing D by 100, we get

19 x + 12 z = 119 -------------------------- E

Similarly substituting equation A in C,

6000 x + 10000 * 2x + 11000 z = 152000

i.e, 26000 x + 11000 z = 152000 ----------------------F

Dividing F by 1000,

26 x + 11 z = 152 ---------------------------G

Now we have two simplified equations,

19 x + 12 z = 119 -------------------------- E and

26 x + 11 z = 152 ---------------------------G

with two equations and two unknowns we can easily solve x and z.

Adding E and G gives,

45 x + 23 z = 271 ---------------------------H

substracting E from G gives,

7x - z = 33 -------------------------------I

From equation I, we have 7x - z = 33

so, z= 7x-33 ---------------- let this be eqn. J

substituting J in H (can be done in E or G also)

45 x + 23 *(7x-33) = 271

45 x + 161 x - 759 = 271

206 x = 1030

So, x = 5

from equation A, we have y=2x

So, y = 10

from equation J, we have z= 7x-33

So, z = 35 - 33 = 2.

Thus Luigi bought five newspaper ads, ten radio commercials and two tv commercials.

2.

This is an optimization problem.

Luigi must get the maximum value out of the function which is the total number of viewers,

6000*x + 10000*y + 11000*z ------------------- A

he can spent maximum of $ 18000

i.e we get the equation,

300*x + 800* y + 1200* z <= 18000

i.e 3x + 8y + 12z <= 180 ------------------- B

Also from the problem data, we have

x <= 5 -------------------------- C

y + z <= 20 ------------------------D

From C and D, we have

x + y + z <= 25 ----------------------------E

So we have two inequalities,

3x + 8y + 12z <= 180 ------------------- B

x + y + z <= 25 ----------------------------E

Multiplying E with 3, we get

3x + 3y + 3z <= 75 -----------------------F

substracting F from B gives,

5y + 9 z <= 105 -------------------G

we have,

y + z <= 20 ------------------------D

multiplying D with 5 gives,

5y + 5 z <= 100 ----------------------H

Now G - H gives,

4z <= 5

i.e z <= 5/4 = 1.25 which says that z can be atmost 1.

Substituting the value of z in D gives,

y <= 19

From equation C,

x<=5

Thus we have (x,y,z) = (5,19,1)

for this combination of (x,y,z), the total cost Luigi has to spent is,

300*5 + 800*19 + 1200*1 = 17,900 which <= $ 18000

And the number of people heard about the sandwich shop will be,

6000*5+10000*19+11000*1 = 2,31,000

Thus, Luigi should buy five newspaper ads, 19 radio commercials and one tv commercial respectively.

Thank you!

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