Joe manufactures cardboard boxes. A small milk company requires boxes in the sha

Question

Joe manufactures cardboard boxes. A small milk company requires boxes in the shape of a rectangular prism to package their products. Joe is contracted to make and deliver boxes for the milk company. For aesthetic reasons the boxes have to be built to certain specifications as follows: The width is three times its depth The height is five times its depth Joe is to deliver 50 boxes per day. a. Find a function that models the volume of the boxes and then determine the dimensions and the volume of the boxes if the depth of a box is 10.5 cm b. Provide a daily estimate for the milk company if cardboard is sold at $5 per square centimetre c. Refine your model to take into account details over-looked in the first estimate d. Generalize your model to provide a daily cost to the milk company e. Make sure the contractor has a simple means of estimating the work

Explanation / Answer

a) Width = 3 * depth = 3 * 10.5 cm = 31.5 cm

Height = 5 * depth = 5 * 10.5 cm = 52.5 cm

Hence the dimensions are 10.5 cm X 31.5 cm X 52.5 cm

Volume of the box = Width * Height * Depth = 10.5 * 31.5 * 52.5 = 17364.375 cm^3

Volume of box (in function of depth) = d * 3d * 5d = 15d^3

b)

Area of rectangular = 2(wd + wh + ld)

=> 2(10.5*31.5+10.5*52.5+31.5*52.5)

=> 5071.5 cm^2

One strange thing the cost $5 per cm^2 is too much since it is too too expensive, but if this is only the data

Then cost of box will be 5071.5 cm^2 * $5/cm^2 = $25357.5

Daily estimate = $25357.5 * 50 = 1.267875 million US$

c) The only issue with the first estimate is that cost was too high, if this is the cost then the cost of cartons will be more than milk selling price leading to the losses

d) Daily Cost = Area of each square box * cost of cardboard per square cm * number of boxes

e) The contractor has the fixed area, so he can make a function of daily cost

Daily cost = Fixed Area * cost of cardboard per square cm * number of boxes

It will help him to estimate the cost if there is a change in cardboard price or the number of boxes which he has to design

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