Trade Show Exhibit Display Booths

Question for anyone thats good at maths?

A manufacturer supplies Display Boxes which are used to protect exhibits at trade shows etc. The boxes have no base, but the walls are made of good quality wood to enhance their experience and the tops are made from a high quality of laminated glass. All the wood used is the same thickness and material. The cost of the boxes depends only on the amount of wood and amount of glass used. The following six boxes (which are shown below with the cost and dimensions) were sent to a museum for an exhibition. My question - How can I find out how much the wood is per sq. ft and how much the glass is per sq. ft. for the following boxes. A £56.20 3ft length, 2ft breadth, 1ft height B £94.60 5.5 length, 2ft breadth, 1ft height C £82.50 6ft length, 2ft breadth, 0.5 ft height D £171.50 5ft length, 4ft breadth, 1.5ft height E £66.30 3ft length, 3ft breadth, 0.75ft height F £127.35 12ft length, 1.5ft breadth, 0.5ft height. Please show all your working so I can learn!

Public Comments

1. If l = length, b = breadth, h = height then the area of wood is perimeter of base * height = 2(l + b) * h =2h(l + b) area of glass = lb Let wood cost £x per sq ft (how come we're using feet and pounds? I thought USA was the only place still using feet!) and glass cost £y per sq ft. then we have A. 2*1*(3+2)x + 2*3y = 56.2 i.e. 10x + 6y = 56.2 B ...15x + 11y = 94.6 C ... 8x + 12y = 82.5 D ... 27x + 20y = 171.5 E ... 9x + 9y = 66.3 F .. 13.5x + 18y = 127.35 Any two of these should give the answers for x and y; if they aren't all satisfied by the same values, there's something wrong. Let's double F: 27x + 36y = 254.7 and subtract D from that: 16y = 83.2 y = 5.2 Sub back into D 27x + 104 = 171.5 27x = 67.5 x = 2.5 Regrettably, these values don't satisfy equation E (69.3, not 66.3) or C (82.4) or B(94.7), but they do give 56.2 for A, So I say the wood costs £2.50 per sq ft, and the glass £5.20, and the costs given for B, C and E should be £94.70, £82.40 and £69.30
2. If the materials used have a constant thickness, their cost varies directly with the surface area. Let w be the cost of 1 ft^2 of wood and Let g be the cost of 1ft^2 of glass. length*breadth=area of glass 2*breadth*height+2*length*height=area of wood So for any given box, cost=(area of wood)w+(area of glass)g You should notice now that you only need two of the six boxes to solve for the two unknowns, g and w. Using boxes A and B, area of wood A=10 ft^2 area of glass A=6 ft^2 area of wood B=15 ft^2 area of glass B=11 ft^2 we have 56.2=10w+6g 94.6=15w+11g multiplying the top equation by 1.5 84.3=15w+9g 94.6=15w+11g subtract the top from the bottom -10.3=-2g g=5.15 and w=2.53 I hope you get it.
3. The glass is L x B, times the cost of the glass The wood is 2H(L + B), times the cost of the wood. Let G be the cost of 1 sq. ft of glass. Let W be the cost of 1 sq. ft of wood. Box A: 6G + 10W = 56.20 Box B: 11G + 15W = 94.60 Box C: 12G + 8W = 82.50 Box D: 20G + 27W = 171.50 Box E: 9G + 9W = 66.30 Box F: 18G + 13.5W = 127.35 Let's take the equations for Box A and Box C: 6G + 10W = 56.20 12G + 8W = 82.50 If you halve the second equation you get: 6G + 10W = 56.20 6G + 4W = 41.25 (half) Now subtract the two equation and 6G will cancel: 6W = 14.95 Divide both sides by 6: W = 14.95 / 6 W = 2.4916666... Plug this back into one of the equations to get G: 6G + 4W = 41.25 6G + 4(2.4916666...) = 41.25 6G = 41.25 - 4(2.4916666...) 6G = 31.28333... G = 31.28333... / 6 G = 5.2138888... Glass is approximately £5.21 / sq. ft. Wood is approximately £2.49 / sq. ft Plugging these back into the box prices we get close to the calculated values: Box A: 6G + 10W = 56.16 (4p under) Box B: 11G + 15W = 94.66 (6p over) Box C: 12G + 8W = 82.44 (6p under) Box D: 20G + 27W = 171.43 (7p under) Box E: 9G + 9W = 69.30 (£3 over) Box F: 18G + 13.5W = 127.395 (4.5p over) Box E seems to be mispriced, the others are pretty close.