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Question
A box with a square base is to have an open top. The surface area of box is 147 sq. cm. What should be its dimensions in order that the volume is largest?
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Solution
Let x cm be the side of square base and h cm be its height.
Then x2 + 4xh = 147
∴ h = `(147 - x^2)/(4x)` ...(1)
Let V = `x^2"h"`
= `x^2((147 - x^2)/(4x))` ...[By (1)]
∴ V = `(1)/(4)(147x - x^3)`
∴ `"dV"/("d"x) = (1)/(4) (147x - x^3) = 0`
∴ 147 = 3x2
∴ `147/3 = x^2`
∴ x2 = 49
∴ x = 7
Put in eq (i)
∴ h = `(147 - x^2)/(4x)`
∴ h = `(147 - 49)/(4(7))`
∴ h = `98/(4 xx 7)`
∴ h = `14/4`
∴ h = `7/2`
∴ h = 3.5
Hence, the volume of the box is largest when the side of square base is 7 cm and its height is 3.5 cm.
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