Question

An open topped box is to be constructed by removing equal squares from
each comer of a 3 metre by 8 metre rectangular sheet of aluminium and folding up the
sides. Find the volume of the largest such box.​

Answer 1

Let the side of the square to be cut off be x cm. Then, the height of the box is x, the length is 3 − 2x, and the breadth is 8 − 2x.

Therefore, the volume V(x) of the box is given by,

V(x) = x(3 - 2x)(8 - 2x)

       = 4x³ - 22x² - 24x

∴ V'(x) = 12x² - 44x + 24

           = 4(3x² - 11x + 6)

           = 4(3x - 2)(x - 3)

∴ V''(x) = 24x - 44

Now,

V'(x) = 0

⇒ x = 2/3, 3

It is not possible to cut off a square of side 3 cm from each corner of the rectangular sheet. Thus, x cannot be equal to 3.

∴x = 2/3

Now,

V''(2/3)

= 24x - 44

= 24(2/3) - 44

= -28 < 0

By second derivative test, x = 2/3 is the point of maxima.

Then:

volume = 4(2/3)³ - 22(2/3)² - 24(2/3)

             = 200/27 cm³

Hope this helps!

Answer 2

Answer:

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