x cm(x + 4) cm© EAGLE BEACON GLOBAL A stationery shop packs notebooks into boxes shaped as shown. Each box has a square base of side \(x\) cm, and the box i...
Question 1 Report
A stationery shop packs notebooks into boxes shaped as shown. Each box has a square base of side \(x\) cm, and the box is 4 cm taller than the side of its base.
- Show that the total surface area of the box, in cm\(^2\), is given by \(S = 6x^{2}+16x\). (2)
- Given that the surface area is \(512\) cm\(^2\), form and solve a quadratic equation to find \(x\). (3)
- State the height of the box for this value of \(x\). (1)
Answer Details
The surface area of this box is the sum of a top, a bottom, and four identical rectangular sides; expressing this total in terms of \(x\) turns a fixed surface area into a quadratic equation for \(x\).
(a) The square top and bottom together contribute \(2x^{2}\). Each of the four sides is \(x\) by \((x+4)\), so the four sides together contribute \(4x(x+4) = 4x^{2}+16x\). Adding these: \(S = 2x^{2}+4x^{2}+16x = 6x^{2}+16x\), as required. [2 marks]
(b) Setting \(S=512\): \(6x^{2}+16x-512=0\), which simplifies (dividing by 2) to \(3x^{2}+8x-256=0\). The discriminant is \(8^{2}-4(3)(-256) = 64+3072 = 3136\), and \(\sqrt{3136}=56\). So \(x = \dfrac{-8\pm56}{6}\), giving \(x=8\) or \(x=-\dfrac{64}{6}\); rejecting the negative root, \(x=8\). [3 marks]
(c) The height is \(x+4 = 8+4 = 12\) cm. [1 mark]
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