Mathematics: Acandy box is made from a piece of cardboard that measures 43 by 23 inches. squares of equal size will be cut out of each corner. the sides will then be folded up to form a rectangular box. what size square should be cut from each corner to obtain maximum volume?

Acandy box is made from a piece of cardboard that

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12.03.2022

For maximum volume of the box, squares with 4.79 inches should be cut off.

Step-by-step explanation:

A candy box is made from a piece of a cardboard that measures 43 × 23 inches.

Let squares of equal size will be cut out of each corner with the measure of x inches.

Therefore, measures of each side of the candy box will become

Length = (43 - 2x)

Width = (23 - 2x)

Height = x

Now we have to calculate the value of x for which volume of the box should be maximum.

Volume (V) = Length×Width×Height

V = (43 -2x)(23 - 2x)(x)

= [(43)×(23) - 46x - 86x + 4x²]x

= [989 - 132x + 4x²]x

= 4x³- 132x² + 989x

Now we find the derivative of V and equate it to 0

$\frac{dV}{dx}=12x^{2}-264x+989$ = 0

Now we get values of x by quadratic formula

$x=\frac{264\pm \sqrt{264^{2}-4\times 12\times 989}}{2\times 12}$

x = 17.212, 4.79

Now we test it by second derivative test for the maximum volume.

$\frac{d"V}{dx}= 24x - 264$

For x = 17.212

$\frac{d"V}{dx}= 24(17.212)-264=413.088-264=149.088$

This value is > 0 so volume will be minimum.

For x = 4.79

$\frac{d"V}{dx}=24(4.79)-264=-149.04$

-149.04 < 0, so volume of the box will be maximum.

Therefore, for x = 4.79 inches volume of the box will be maximum.

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