Shopping malls

The chain of department stores plans to invest up to 24,000 euros in television advertising. All commercials will be placed on a television station where the broadcast of a 30-second spot costs EUR 1,000 and is watched by 14,000 potential customers, during prime time it costs EUR 2,000 and is watched by 24,000 potential customers, and late at night it costs EUR 1,500 and is watched 18 000 potential customers. The TV station will not accept an order to broadcast more than 15 spots in total at all three times. How many spots should the chain order at each time to maximize the total number of viewers who will watch them? How many potential customers will watch the spots?

Correct result:

n =  296000

Solution:

$a \cdot \ 1000+b \cdot \ 2000+c \cdot \ 1500 <=24000 \ \\ a+b+c < 15 \ \\ n=14000 \cdot \ a+24000 \cdot \ b+18000 \cdot \ c \ \\ \ \\ max n \ \\ \ \\ a_{1}=0; b_{1}=0; c_{1}=7 \ \\ \ \\ d_{1}=a_{1} \cdot \ 1000+b_{1} \cdot \ 2000+c_{1} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+7 \cdot \ 1500=10500 \ \\ \ \\ n_{1}=14000 \cdot \ a_{1}+24000 \cdot \ b_{1}+18000 \cdot \ c_{1}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 7=126000 \ \\ \ \\ a_{2}=0; b_{2}=0; c_{2}=8 \ \\ \ \\ d_{2}=a_{2} \cdot \ 1000+b_{2} \cdot \ 2000+c_{2} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+8 \cdot \ 1500=12000 \ \\ \ \\ n_{2}=14000 \cdot \ a_{2}+24000 \cdot \ b_{2}+18000 \cdot \ c_{2}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 8=144000 \ \\ \ \\ a_{3}=0; b_{3}=0; c_{3}=9 \ \\ \ \\ d_{3}=a_{3} \cdot \ 1000+b_{3} \cdot \ 2000+c_{3} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+9 \cdot \ 1500=13500 \ \\ \ \\ n_{3}=14000 \cdot \ a_{3}+24000 \cdot \ b_{3}+18000 \cdot \ c_{3}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 9=162000 \ \\ \ \\ a_{4}=0; b_{4}=0; c_{4}=10 \ \\ \ \\ d_{4}=a_{4} \cdot \ 1000+b_{4} \cdot \ 2000+c_{4} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+10 \cdot \ 1500=15000 \ \\ \ \\ n_{4}=14000 \cdot \ a_{4}+24000 \cdot \ b_{4}+18000 \cdot \ c_{4}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 10=180000 \ \\ \ \\ a_{5}=0; b_{5}=0; c_{5}=11 \ \\ \ \\ d_{5}=a_{5} \cdot \ 1000+b_{5} \cdot \ 2000+c_{5} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+11 \cdot \ 1500=16500 \ \\ \ \\ n_{5}=14000 \cdot \ a_{5}+24000 \cdot \ b_{5}+18000 \cdot \ c_{5}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 11=198000 \ \\ \ \\ a_{6}=0; b_{6}=0; c_{6}=12 \ \\ \ \\ d_{6}=a_{6} \cdot \ 1000+b_{6} \cdot \ 2000+c_{6} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+12 \cdot \ 1500=18000 \ \\ \ \\ n_{6}=14000 \cdot \ a_{6}+24000 \cdot \ b_{6}+18000 \cdot \ c_{6}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 12=216000 \ \\ \ \\ a_{7}=0; b_{7}=0; c_{7}=13 \ \\ \ \\ d_{7}=a_{7} \cdot \ 1000+b_{7} \cdot \ 2000+c_{7} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+13 \cdot \ 1500=19500 \ \\ \ \\ n_{7}=14000 \cdot \ a_{7}+24000 \cdot \ b_{7}+18000 \cdot \ c_{7}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 13=234000 \ \\ \ \\ a_{8}=0; b_{8}=0; c_{8}=14 \ \\ \ \\ d_{8}=a_{8} \cdot \ 1000+b_{8} \cdot \ 2000+c_{8} \cdot \ 1500=0 \cdot \ 1000+0 \cdot \ 2000+14 \cdot \ 1500=21000 \ \\ \ \\ n_{8}=14000 \cdot \ a_{8}+24000 \cdot \ b_{8}+18000 \cdot \ c_{8}=14000 \cdot \ 0+24000 \cdot \ 0+18000 \cdot \ 14=252000 \ \\ \ \\ a_{9}=0; b_{9}=1; c_{9}=13 \ \\ \ \\ d_{9}=a_{9} \cdot \ 1000+b_{9} \cdot \ 2000+c_{9} \cdot \ 1500=0 \cdot \ 1000+1 \cdot \ 2000+13 \cdot \ 1500=21500 \ \\ \ \\ n_{9}=14000 \cdot \ a_{9}+24000 \cdot \ b_{9}+18000 \cdot \ c_{9}=14000 \cdot \ 0+24000 \cdot \ 1+18000 \cdot \ 13=258000 \ \\ \ \\ a_{10}=0; b_{10}=2; c_{10}=12 \ \\ \ \\ d_{10}=a_{10} \cdot \ 1000+b_{10} \cdot \ 2000+c_{10} \cdot \ 1500=0 \cdot \ 1000+2 \cdot \ 2000+12 \cdot \ 1500=22000 \ \\ \ \\ n_{10}=14000 \cdot \ a_{10}+24000 \cdot \ b_{10}+18000 \cdot \ c_{10}=14000 \cdot \ 0+24000 \cdot \ 2+18000 \cdot \ 12=264000 \ \\ \ \\ a_{11}=0; b_{11}=3; c_{11}=11 \ \\ \ \\ d_{11}=a_{11} \cdot \ 1000+b_{11} \cdot \ 2000+c_{11} \cdot \ 1500=0 \cdot \ 1000+3 \cdot \ 2000+11 \cdot \ 1500=22500 \ \\ \ \\ n_{11}=14000 \cdot \ a_{11}+24000 \cdot \ b_{11}+18000 \cdot \ c_{11}=14000 \cdot \ 0+24000 \cdot \ 3+18000 \cdot \ 11=270000 \ \\ \ \\ a_{12}=0; b_{12}=4; c_{12}=10 \ \\ \ \\ d_{12}=a_{12} \cdot \ 1000+b_{12} \cdot \ 2000+c_{12} \cdot \ 1500=0 \cdot \ 1000+4 \cdot \ 2000+10 \cdot \ 1500=23000 \ \\ \ \\ n_{12}=14000 \cdot \ a_{12}+24000 \cdot \ b_{12}+18000 \cdot \ c_{12}=14000 \cdot \ 0+24000 \cdot \ 4+18000 \cdot \ 10=276000 \ \\ \ \\ a_{13}=0; b_{13}=5; c_{13}=9 \ \\ \ \\ d_{13}=a_{13} \cdot \ 1000+b_{13} \cdot \ 2000+c_{13} \cdot \ 1500=0 \cdot \ 1000+5 \cdot \ 2000+9 \cdot \ 1500=23500 \ \\ \ \\ n_{13}=14000 \cdot \ a_{13}+24000 \cdot \ b_{13}+18000 \cdot \ c_{13}=14000 \cdot \ 0+24000 \cdot \ 5+18000 \cdot \ 9=282000 \ \\ \ \\ a_{14}=0; b_{14}=6; c_{14}=8 \ \\ \ \\ d_{14}=a_{14} \cdot \ 1000+b_{14} \cdot \ 2000+c_{14} \cdot \ 1500=0 \cdot \ 1000+6 \cdot \ 2000+8 \cdot \ 1500=24000 \ \\ \ \\ n_{14}=14000 \cdot \ a_{14}+24000 \cdot \ b_{14}+18000 \cdot \ c_{14}=14000 \cdot \ 0+24000 \cdot \ 6+18000 \cdot \ 8=288000 \ \\ \ \\ a_{15}=1; b_{15}=7; c_{15}=6 \ \\ \ \\ d_{15}=a_{15} \cdot \ 1000+b_{15} \cdot \ 2000+c_{15} \cdot \ 1500=1 \cdot \ 1000+7 \cdot \ 2000+6 \cdot \ 1500=24000 \ \\ \ \\ n_{15}=14000 \cdot \ a_{15}+24000 \cdot \ b_{15}+18000 \cdot \ c_{15}=14000 \cdot \ 1+24000 \cdot \ 7+18000 \cdot \ 6=290000 \ \\ \ \\ a_{16}=2; b_{16}=8; c_{16}=4 \ \\ \ \\ d_{16}=a_{16} \cdot \ 1000+b_{16} \cdot \ 2000+c_{16} \cdot \ 1500=2 \cdot \ 1000+8 \cdot \ 2000+4 \cdot \ 1500=24000 \ \\ \ \\ n_{16}=14000 \cdot \ a_{16}+24000 \cdot \ b_{16}+18000 \cdot \ c_{16}=14000 \cdot \ 2+24000 \cdot \ 8+18000 \cdot \ 4=292000 \ \\ \ \\ a_{17}=3; b_{17}=9; c_{17}=2 \ \\ \ \\ d_{17}=a_{17} \cdot \ 1000+b_{17} \cdot \ 2000+c_{17} \cdot \ 1500=3 \cdot \ 1000+9 \cdot \ 2000+2 \cdot \ 1500=24000 \ \\ \ \\ n_{17}=14000 \cdot \ a_{17}+24000 \cdot \ b_{17}+18000 \cdot \ c_{17}=14000 \cdot \ 3+24000 \cdot \ 9+18000 \cdot \ 2=294000 \ \\ \ \\ a_{18}=4; b_{18}=10; c_{18}=0 \ \\ \ \\ d_{18}=a_{18} \cdot \ 1000+b_{18} \cdot \ 2000+c_{18} \cdot \ 1500=4 \cdot \ 1000+10 \cdot \ 2000+0 \cdot \ 1500=24000 \ \\ \ \\ n_{18}=14000 \cdot \ a_{18}+24000 \cdot \ b_{18}+18000 \cdot \ c_{18}=14000 \cdot \ 4+24000 \cdot \ 10+18000 \cdot \ 0=296000 \ \\ \ \\ n=296000$

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