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Containers- Minimum area An Investigation

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Containers Consider the container It is required to design a box which satisfies the following requirements:- a) The box is to hold 2000cm 3 (2 litres) b) The length of the box is to be two times the breadth of the box. c) The amount of cardboard needed to make the box is to be a minimum. x 2x h

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Steps:- 1. Use the fact that V = LBH to show that 2 Show that the total area of cardboard needed is 3. Put into Y 1 and use the TABLE function on the graphic calculator to obtain the solution.

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Table 1 Complete the table shown to show your results. XArea (cm 2 ) 16004 23016 3 4 5 6 7 8 9 10 11 12

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You can see from the table that the minimum is somewhere between x = 8 and x = 10. So we now form another table with x going up in steps of 0.2 say. On your calculator set TblStart to 8 and ∆Tbl to 0.2 and then complete the Table 2 on the next slide.

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Table 2 Why have we stopped this Table at 9.6? XArea (cm 2 ) 81006 8.21000.7 8.4 8.6 8.8 9 9.2 9.4 9.6

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Make up a third table starting at ? and going up in steps of 0.1 to obtain the final value for x which will give the minimum area of cardboard. Now that you have this value of x, draw a box like the one below and mark on the box the dimensions which the box would have for this minimum area.

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Note A company using this box for packaging goods, would produce millions of these boxes and the cost of producing them would be a minimum because each box uses the smallest amount of cardboard and satisfies the other conditions stated initially in the problem. Clean It Washing Powder

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Container Examples (Using the graphic calculator Table function) 1. The container is to hold a volume of 800cm 3. The length is to be five times the breadth. The area of cardboard needed to make the container is to be a minimum. Find x, then draw the container. h 5x x

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Ex2. Design a container to satisfy the following requirements. The container is to hold a volume of 1200cm 3. The container has a square base. The area of cardboard needed to make the container is to be a minimum. h x x

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Ex3. Design the container below The container is to hold a volume of 1500cm 3. The length is to be two times the breadth. The area of cardboard needed to make the container is to be a minimum. h x 2x

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Ex.4 This container is to hold a volume of 1600cm 3. The length is to be three times the breadth. The area of cardboard needed to make the container is to be a minimum. h 3x x

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