. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 2323 ft by 1313 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way. b. Suppose that in part (a) the original piece of cardboard is a square with sides of length s. Find the volume of the largest box that can be formed in this way.

The volume of a box is the amount of space in it.

The dimension of the cardboard is given as:

Assume the cut-out is x.

So, the dimension of the box is:

The volume of the box is:

Expand

Differentiate

Set to 0

Rewrite as:

Using a calculator, we have:

9.33 is greater than the dimensions of the box.

So, the possible value of x is:

Recall that:

So, we have:

Hence, the maximum value of the box is

If the dimension of the cardboard is s,

The dimension of the box would be:

So, the volume is:

Substitute

Expand

Integrate

Set to 0

Using a calculator, we have:

5.34 is greater than the side lengths. So, we have:

Recall that:

So, we have:

Hence, the volume of the largest box is

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(a)Therefore the volume of the box is 361.19 cubic ft.

(b)Therefore the volume of the box is 361.19 cubic ft.

Step-by-step explanation:

Given that,

Squares with sides length x are cut out from of each corner of a rectangular piece of cardboard measuring 23 ft and 13 ft.

Now the length of the box is =(23-2x) ft

The width of the box is =(13-2x) ft

The height of the box is= x

The volume of the box is = Length×width×Height

                                          =(23-2x)(13-2x)x cubic ft

                                         = 299x-72x²+4x³ cubic ft

Let,

V=299x-72x²+4x³

Differentiating with respect to x

V'= 299-144x+12x²

Again differentiating with respect to x

V''= -144+24x

To find the maximum volume, we set V'=0

∴299-144x+12x²=0

Applying quadratic formula , here a=12,b= -144 and c=299

For x= 9.33 , the width of the box will negative which is impossible.

So, x= 2.67

So, at x=2.67, the volume of the box will be maximum.

Therefore the volume of the box is

=(299x-72x²+4x³) cubic ft

=361.19 cubic ft

(b)

The value of s will same with the value of x

so the volume of the box also remains same.

Therefore the volume of the box is 361.19 cubic ft

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step-by-step explanation: