Shopping malls

The chain of department stores plans to invest up to 24,000 euros in television advertising. Ads will place all commercials 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. Twenty-four thousand potential customers watch it, and it costs EUR 1,500 late at night and is watched by 18,000 potential customers. The TV station will not accept an order to broadcast more than 15 spots in total three times. How many spots should the chain order each time to maximize the total number of viewers watching them? How many potential customers will watch the spots?

Correct answer:

n =  296000

Step-by-step explanation:

$$\begin{gathered} a\cdot 1000+b\cdot 2000+c\cdot 1500\le 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 \end{gathered}$$

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