a contractor wants to bid on an order to make 100,000 boxes from cardboard that

Question

a contractor wants to bid on an order to make 100,000 boxes from cardboard that costs $.08 per square foot. Each box must have a square base and volume of 4 cubic feet. The sides of the box will be made from a single piece of cardboard, folded three times and taped at the seam.
The top and bottom will be made from separate pieces of cardboard with the bottom taped
on all four sides and the top taped on just one edge to create a lid that opens. Taping costs $.03 per foot.

a. Find the dimensions of the box of minimum cost.
b. How much should the contractor bid (in dollars) to make a profit of 17%?
c. How much should the bid in part b be increased if the costumer insists on cubical boxes?

Explanation / Answer

Volume = Area of Base x height = 4

Let x be equal to the sides of the base, and h equal the height of the box

Then,

V = x2h = 4 -----> h = 4/(x2)

Since the sides of the box are made from a single sheet and folded 3 times(to make 4 sides), its dimensions are:

4x by h ----> A = 4xh

The top and bottom each have dimensions of x by x (since the box must have a square base)

A = x2 + x2 = 2x2

Therefore, the total surface area of the box is:

SA = 4xh + 2x2

SA = (4/x) + 2x2

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a.)

Cost of box:

C = (0.08)SA + (0.03)(h + 4x + x) = (0.08)( (4/x) + 2x2 ) + (0.03)(h + 5x)

C = (0.08)( (4/x) + 2x2 ) + (0.03)( 4/(x2) + 5x)

Min Cost ---> set derivative of cost to 0

0 = C ' = 0.15 - (0.24/x3) - (0.32/x2) + 0.32x

Solving this, we get x = 1.058 -----> h = 3.573

The dimensions of the box that minimize cost are:

1.058 ft by 1.058 ft by 3.573 ft

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b.)

Profit = Revenue - Cost

Profit = 17% -----> Revenue = 1.17 x Cost

Minimum Cost =  (0.08)( (4/x) + 2x2 ) + (0.03)( 4/(x2) + 5x) where x = 1.058

Minimum Cost = 0.747 ----> Revenue = 0.874

Since 100,000 boxes have to be made:

Bid = 100,000 x Revenue -----> Bid = $87,400

The contractor should bid $87,400 for a profit of 17%

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c.)

Cubical Box:

V = x3 = 4

x = 1.587

SA (for a cube) = 6x2

SA = 15.119

Cost:

C = (0.08)(SA) + (0.03)(6x)

C = 1.49526

Profit = Revenue - Cost

Profit = 17% ----> Revenue = 1.17 x Cost

Cost = 1.495 ----> Revenue = 1.74945

Bid = 100,000 x Revenue

Bid = $174,945

Change in Bid: $174,945 - $87,400 = $87,545

The Bid would be increased by $87,545 for cubical boxes

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