squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 19 ft by 11 ft. the resu

Question

9 months ago

15

lting 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.

1 answer:

finlep [7] · Answer 1 · 2024-04-15T10:21:33+03:00

The maximum value of the box is 361.186in³

Given,

In the question:

The dimension of the cardboard is given as:

Length of the cardboard is = 23

Width of the cardboard is = 13

and, squares with sides of length x

To find the volume of the largest box that can be formed in this way.

Now, According to the question:

Assume the cut-out is x.

So, the dimension of the box is:

Length = 23

Width = 13

Height = x

The volume of the box will be:

V = (23 - 2x ) (13 - 2x)x

V = 299 - 72x² + 4x³

Differentiate w.r.t to x

V = 299 - 144x + 12x²

Set to 0

V = 299 - 144x + 299 = 0

we have:

x = (2.67, 9.33)

9.33 is greater than the dimensions of the box.

So, the possible value of x is:

x = 2.67

Recall that:

V = 299 - 72x² + 4x³

So, we have:

V = 299 (2.67) - 72 (2.67)² + 4(2.67)³

Hence, the maximum value of the box is 361.186in³

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