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 1000+b 2000+c 1500<=24000 a+b+c<15 n=14000 a+24000 b+18000 c  maxn  a1=0;b1=0;c1=7  d1=a1 1000+b1 2000+c1 1500=0 1000+0 2000+7 1500=10500  n1=14000 a1+24000 b1+18000 c1=14000 0+24000 0+18000 7=126000  a2=0;b2=0;c2=8  d2=a2 1000+b2 2000+c2 1500=0 1000+0 2000+8 1500=12000  n2=14000 a2+24000 b2+18000 c2=14000 0+24000 0+18000 8=144000  a3=0;b3=0;c3=9  d3=a3 1000+b3 2000+c3 1500=0 1000+0 2000+9 1500=13500  n3=14000 a3+24000 b3+18000 c3=14000 0+24000 0+18000 9=162000  a4=0;b4=0;c4=10  d4=a4 1000+b4 2000+c4 1500=0 1000+0 2000+10 1500=15000  n4=14000 a4+24000 b4+18000 c4=14000 0+24000 0+18000 10=180000  a5=0;b5=0;c5=11  d5=a5 1000+b5 2000+c5 1500=0 1000+0 2000+11 1500=16500  n5=14000 a5+24000 b5+18000 c5=14000 0+24000 0+18000 11=198000  a6=0;b6=0;c6=12  d6=a6 1000+b6 2000+c6 1500=0 1000+0 2000+12 1500=18000  n6=14000 a6+24000 b6+18000 c6=14000 0+24000 0+18000 12=216000  a7=0;b7=0;c7=13  d7=a7 1000+b7 2000+c7 1500=0 1000+0 2000+13 1500=19500  n7=14000 a7+24000 b7+18000 c7=14000 0+24000 0+18000 13=234000  a8=0;b8=0;c8=14  d8=a8 1000+b8 2000+c8 1500=0 1000+0 2000+14 1500=21000  n8=14000 a8+24000 b8+18000 c8=14000 0+24000 0+18000 14=252000  a9=0;b9=1;c9=13  d9=a9 1000+b9 2000+c9 1500=0 1000+1 2000+13 1500=21500  n9=14000 a9+24000 b9+18000 c9=14000 0+24000 1+18000 13=258000  a10=0;b10=2;c10=12  d10=a10 1000+b10 2000+c10 1500=0 1000+2 2000+12 1500=22000  n10=14000 a10+24000 b10+18000 c10=14000 0+24000 2+18000 12=264000  a11=0;b11=3;c11=11  d11=a11 1000+b11 2000+c11 1500=0 1000+3 2000+11 1500=22500  n11=14000 a11+24000 b11+18000 c11=14000 0+24000 3+18000 11=270000  a12=0;b12=4;c12=10  d12=a12 1000+b12 2000+c12 1500=0 1000+4 2000+10 1500=23000  n12=14000 a12+24000 b12+18000 c12=14000 0+24000 4+18000 10=276000  a13=0;b13=5;c13=9  d13=a13 1000+b13 2000+c13 1500=0 1000+5 2000+9 1500=23500  n13=14000 a13+24000 b13+18000 c13=14000 0+24000 5+18000 9=282000  a14=0;b14=6;c14=8  d14=a14 1000+b14 2000+c14 1500=0 1000+6 2000+8 1500=24000  n14=14000 a14+24000 b14+18000 c14=14000 0+24000 6+18000 8=288000  a15=1;b15=7;c15=6  d15=a15 1000+b15 2000+c15 1500=1 1000+7 2000+6 1500=24000  n15=14000 a15+24000 b15+18000 c15=14000 1+24000 7+18000 6=290000  a16=2;b16=8;c16=4  d16=a16 1000+b16 2000+c16 1500=2 1000+8 2000+4 1500=24000  n16=14000 a16+24000 b16+18000 c16=14000 2+24000 8+18000 4=292000  a17=3;b17=9;c17=2  d17=a17 1000+b17 2000+c17 1500=3 1000+9 2000+2 1500=24000  n17=14000 a17+24000 b17+18000 c17=14000 3+24000 9+18000 2=294000  a18=4;b18=10;c18=0  d18=a18 1000+b18 2000+c18 1500=4 1000+10 2000+0 1500=24000  n18=14000 a18+24000 b18+18000 c18=14000 4+24000 10+18000 0=296000  n=296000a \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



We would be pleased if you find an error in the word problem, spelling mistakes, or inaccuracies and send it to us. Thank you!






Showing 0 comments:
avatar




Tips to related online calculators
Do you want to convert time units like minutes to seconds?

You need to know the following knowledge to solve this word math problem:

Next similar math problems:

  • Ascend vs. descent
    lines Which function is growing? a) y = 2-x b) y = 20 c) y = (x + 2). (-5) d) y = x-2
  • Secret treasure
    max_cylinder_pyramid Scouts have a tent in the shape of a regular quadrilateral pyramid with a side of the base 4 m and a height of 3 m. Determine the radius r (and height h) of the container so that they can hide the largest possible treasure.
  • Rectangle pool
    basen_5 Determine dimensions of open pool with a square bottom with a capacity 32 m3 to have painted/bricked walls with least amount of material.
  • Minimum of sum
    derive_1 Find a positive number that the sum of the number and its inverted value was minimal.
  • Paper box
    box Hard rectangular paper has dimensions of 60 cm and 28 cm. The corners are cut off equal squares and the residue was bent to form an open box. How long must be side of the squares to be the largest volume of the box?
  • Goat
    koza Meadow is a circle with radius r = 19 m. How long must a rope to tie a goat to the pin on the perimeter of the meadow to allow goat eat half of meadow?
  • Sphere in cone
    sphere-in-cone A sphere of radius 3 cm desribe cone with minimum volume. Determine cone dimensions.
  • Cone
    diag22 Into rotating cone with dimensions r = 8 cm and h = 8 cm incribe cylinder with maximum volume so that the cylinder axis is perpendicular to the axis of the cone. Determine the dimensions of the cylinder.
  • Sphere and cone
    cone_in_sphere Within the sphere of radius G = 33 cm inscribe cone with largest volume. What is that volume and what are the dimensions of the cone?
  • Derivation
    fx Exists a function whose derivation is the same function?
  • Fall
    rocket The body was thrown vertically upward at speed v0 = 79 m/s. Body height versus time describe equation ?. What is the maximum height body reach?
  • Statue
    michelangelo On the pedestal high 4 m is statue 2.7 m high. At what distance from the statue must observer stand to see it in maximum viewing angle? Distance from the eye of the observer from the ground is 1.7 m.
  • Ladder
    rebrik_4 4 m long ladder touches the cube 1mx1m at the wall. How high reach on the wall?
  • Cylindrical container
    valec2_6 An open-topped cylindrical container has a volume of V = 3140 cm3. Find the cylinder dimensions (radius of base r, height v) so that the least material is needed to form the container.
  • Curve and line
    parabol The equation of a curve C is y=2x² -8x+9 and the equation of a line L is x+ y=3 (1) Find the x co-ordinates of the points of intersection of L and C. (2) Show that one of these points is also the stationary point of C?
  • Derivative problem
    derive The sum of two numbers is 12. Find these numbers if: a) The sum of their third powers is minimal. b) The product of one with the cube of the other is maximal. c) Both are positive and the product of one with the other power of the other is maximal.
  • The position
    motion2_3 The position of a body at any time T is given by the displacement function S=t3-2t2-4t-8. Find its acceleration at each instant time when the velocity is zero.