Open box of a given volume
Open box of a given volume
IP 3
8 marks
An open box is made from a rectangular sheet of card \(12\) cm by \(8\) cm by cutting a square of side \(x\) cm from each corner and folding up the sides.
- Show that the volume, \(V\) cm\(^3\), of the box is given by \(V=4x^3-40x^2+96x\), and state the range of values of \(x\) for which the model is valid.[3]
- Given that the volume is \(64\) cm\(^3\), show that \(x=2\) is one possible value and find, in exact form, the other values of \(x\) satisfying the equation.[4]
- Explain which of your values in (b) give a box with volume \(64\) cm\(^3\).[1]
Solution:
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Answer:(a) \(0<x<4\) (b) \(x=4\pm2\sqrt{2}\) (c) \(x=2\) or \(x=4-2\sqrt{2}\)
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