A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in ) is equal to [2023]

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Solution

Q.
A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in ${cm}^{2}$ ) is equal to [2023]

1 675

2 900

3 1025

4 800

Ans.
(4)

Volume of cuboid, $v (x) = {(30 - 2 x)}^{2} x$

$∴$    $\frac{d v}{d x} = {(30 - 2 x)}^{2} + 2 x (30 - 2 x) (- 2) = 0$

$\Rightarrow (30 - 2 x) (30 - 2 x - 4 x) = 0$

$\Rightarrow (30 - 2 x) (30 - 6 x) = 0 \Rightarrow x = 5, 15$

$∵$    $x = 15 not possible,$    $∴$ $x = 5$

$\frac{d^{2} v}{d x^{2}} = 24 x - 240$

$At x = 5; \frac{d^{2} v}{d x^{2}} = 24 (5) - 240 = - 120 < 0$

$∴$    $The volume of the box is maximum at x = 5$

$∴$ Surface area $= {(30 - 2 x)}^{2} + 4 x (30 - 2 x)$

$= {(20)}^{2} + 4 \times 5 \times 20 = 400 + 400 = 800 {cm}^{2}$

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