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A box with an open top is to be constructed from a 15 inch by 24 inch piece of cardboard by cutting squares of equal sides from the corners and folding up the sides. Find the dimensions of the box of the largest volume that can be constructed.

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okpalawalter8

The dimensions of the box of the largest volume that can be constructed are

D=18,9,3

What are the dimensions?

Generally, the equation for the lenght is  mathematically given as

Lenght=(24-2x)

Widht= (15-2x)

Therefore

Volume =L*B*H

V=(24-2x)(15-2x)x

V=(360-48x-30x+4x)x

V=4x^3-78x^2+360x

Differnciating we have

dv/dx=12x^2-156x+360

put dv/dx=0

12(x^2-13x+30)=0

x^2-10x-3x+30=0

(x-10)(x-3)=0

root=10,3

Second diiferciation

d^v/dx^2=24x-156

[tex]lim _{x=3} =-84 < 0[/tex]

In conclusion,

L=24-2x

L=24-2x3

L=18in

w= 15-2x

w=15-2*3

w=9in

h=x

h=2in

Dimension=18,9,3

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