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You are creating an open top box with a piece of cardboard that is 16 x 30“. What size of square should be cut out of each corner to create a box with the largest volume?

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Answer:


(10)/(3) \ inches of square should be cut out of each corner to create a box with the largest volume.

Explanation:

Given: Dimension of cardboard= 16 x 30“.

As per the dimension given, we know Lenght is 30 inches and width is 16 inches. Also the cardboard has 4 corners which should be cut out.

Lets assume the cut out size of each corner be "x".

∴ Size of cardboard after 4 corner will be cut out is:

Length (l)=
30-2x

Width (w)=
16-2x

Height (h)=
x

Now, finding the volume of box after 4 corner been cut out.

Formula; Volume (v)=
l* w* h

Volume(v)=
(30-2x)* (16-2x)* x

Using distributive property of multiplication

⇒ Volume(v)=
4x^(3) -92x^(2) +480x

Next using differentiative method to find box largest volume, we will have
(dv)/(dx)= 0


(d (4x^(3) -92x^(2) +480x))/(dx) = (dv)/(dx)

Differentiating the value


(dv)/(dx) = 12x^(2) -184x+480

taking out 12 as common in the equation and subtituting the value.


0= 12(x^(2) -(46x)/(3) +40)

solving quadratic equation inside the parenthesis.


12(x^(2) -12x-(10x)/(x) +40)=0

Dividing 12 on both side


[x(x-12)-(10)/(3) (x-12)]= 0

We can again take common as (x-12).


x(x-12)[x-(10)/(3) ]=0


(x-(10)/(3) ) (x-12)= 0

We have two value for x, which is
12 and (10)/(3)

12 is invalid as, w=
(16-2x)= 16-2* 12

∴ 24 inches can not be cut out of 16 inches width.

Hence, the cut out size from cardboard is
(10)/(3)\ inches

Now, subtituting the value of x to find volume of the box.

Volume(v)=
(30-2x)* (16-2x)* x

⇒ Volume(v)=
(30-2* (10)/(3) )* (16-2* (10)/(3))* (10)/(3)

⇒ Volume(v)=
(30-(20)/(3) ) (16-(20)/(3)) ((10)/(3) )

Volume(v)= 725.93 inches³

User Laquanda
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