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mestny [16]
1 year ago
6

A rectangular sheet of tin measures 20 inches by 12 inches. suppose you cut a square out of each corner and fold up the sides to

make an open-topped box. what size square should you cut out to maximize the box's volume? show all work to earn credit.
Mathematics
1 answer:
stich3 [128]1 year ago
8 0

The volume of the box is maximized when a 6 x 6 inch or 10 x 10 inch square is cut from the corners of the rectangular sheet.

Volume of rectangle:

The volume of an object is the amount of space occupied by the object or shape, which is in three-dimensional space. It is usually measured in terms of cubic units.

The formula for Volume of the rectangle is

V = l x b x h

where

l represents the length

b represents the breadth

h represents the height.

Given,

A rectangular sheet of tin measures 20 inches by 12 inches. suppose you cut a square out of each corner and fold up the sides to make an open-topped box.

Here we need to find the volume of the box is maximized when a square is cut from the corners of the rectangular sheet.

Here we have the following values:

h = x

l = 20 - 2x

b = 12 - 2x.

Here we have to subtract two times of x value because in the question they said that we have to cut a square from it.

Apply the values on the volume formula then we get,

=> V = (20 - 2x) x (12 - 2x) x (x)

=> V = [240 - 40x - 24x + 4x²] x (x)

=> V = [240x - 64x² + 4x³]

Now differentiate:

=> V' =  240 - 64x + 4x²

Simplify and order the equation,

Then we get,

=> V' = x² - 16x + 60

When we factorize the equation then we get,

=> V' = x² - 6x - 10x + 60

Take the common term out of it,

=> V' = x (x-6) -10(x -6)

Therefore, the value of x is, either 6 or 10.

Now you need to look at your solutions.

At x=6, the volume is,

=> V = (20 - 2(6)) x (12 - 2(6)) x (6)

=> V = (20 - 12) x (12 - 12) x 6

=> V = 8  x 0 x 6

=> V = 0

our corner squares devour the entire piece of cardboard and you are left with zero volume; so that solution is a minimum.

Now at the value of x = 10,

The volume is

=> V = (20 - 2(10) x (12 - 2(10)) x (10)

=> V = (20 - 20) x (12 - 20) x 10

=> V = 0 x (-8) x 10

=> V = 0

So, in both cases we have 0 as minimum volume. So, we can take any one of the value to get the maximum out of it.

To know more about Volume of the Rectangle here.

brainly.com/question/13798973

#SPJ4

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