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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?
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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