A square sheet of cardboard with each side 'a' cm is to be used to make an open top box by cutting a small square of cardboard from each of the corners and bending up the sides.
Answers
Explanation:
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Answer - 432cm³
Given, A square sheet of cardboard with each side 'a' cm is to be used to make an open top box
To find ,
The maximum volume of the box.
Explanation -
Let the square's sides that are severed from each corner measure x cm.
The box's height will then be x cm and its base will have sides of 18–2x cm.
Consequently, box volume
V = (182x)(182x)x V = (18-2x)
²x V = 4x³ +324x - 72x² ... I
When we differentiate w.rt to x,
we obtain dV dx dV dx.
In the case of maximal volume,
12(x2 - 12x + 27) = 0;
x2 - 9x -3x+27; 0;
(x-9)(x-3); 0; and x = 9,
3 = 12x2 + 324 - 144x 12(x2 - 12x + 27)... (ii)
differentiating, we get
= 2x - 12 ...(iii)
At x = 9,
...V is minimum at x = 9 at x = 3
= +ve
...V is maximum at x = 3
... Maximum volume V = (18-6)(18-6) × 3
= 12 × 12 × 3 = 432cm³