An open-top box is to be made from a 42-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side. What should be the length of the side x of the square cut out of each corner to get a box with the maximum volume

x  = 7.45 inch for the maximum volume.

The area of the square = x² = 7.45² = 55.5025  inch²

Step-by-step explanation:

From the given information:

An open-top box is to be made from a 42-inch by 48-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side.

The objective is to determine the length of the side x of the square cut out of each corner to get a box with the maximum volume

The volume of the box = l×b×h

The volume of the box =

The volume of the box =

The volume of the box =

The volume of the box =

The volume of the box =

For the maximum volume V' = 0

V' =

Using the quadratic formula; we have:

where;

a = 12 , b = -360 c = 2016

For the maximum value , we check the points in the second derivative term

V'' = 24x - 360

V'' ( 22.55) = 24(22.55) - 360

V'' ( 22.55) = 541.2 - 360  

V'' ( 22.55) = 181.2   (minimum)

V'' ( 7.45) = 24(7.45) - 360

V'' ( 7.45) = 178.8 - 360  

V'' ( 7.45) = -181.2  < 0  (maximum)

Therefore, x  = 7.45 inch for the maximum volume.

The area of the square = x² = 7.45² = 55.5025  inch²

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step-by-step explanation: ; )