What is the maximum volume in cubic inches of an open box to be made from a 12-inch by 16-inch piece of cardboard by cutting out squares of equal sides from the four corners and bending up the sides? your work must include a statement of the function and its derivative. give one decimal place in your final answer. (10 points)

194.07 cubic inches.

Step-by-step explanation:

The cardboard is 12x16 before removing a square from each end.  This square is x inches wide.  Thus the 16 inside is shortened by x inches on both sides, or it is now 16-2x inches.  The 12 inside is also reduced by 2x.  The x value is also the height of the box when you fold the sides up.  Thus the volume V = wlh = (16-2x)*(12-2x)*(x) = 4x^3 - 56x^2 + 192x.  

To find the maximum, take the derivative, and find its roots  

V = 4x^3 - 56x^2 + 192x

dV/dx = 12x^2 - 112x + 192

The roots are (14+2(13)^.5)/3 ~= 7.07 and (14-2(13)^.5)/3 ~= 2.26  

The roots would be the possible values of x, the square we cut.  Since 7.07 x 2 = 14.14 inches, this exceeds the 12 inch side, thus x = 2.26 inches.  Thus you cut 2.26 inches from each corner to obtain the maximum volume.  

Cube is 11.48 x 7.48 x 2.26 with a volume of 194.07 cubic inches.

If you start with a 12x16 rectangle and cut square with side length x, when you bend the sides you'll have an inner rectangle with sides and , and a height of x.

So, the volume will be given by the product of the dimensions, i.e.

The derivative of this function is

and it equals zero if and only if

If we evaluate the volume function at these points, we have

So, the maximum volume is given if you cut a square with side length

answer: 48

just remember to use pemdas:

parentheses

expressions

multiplication

division

addition

subtraction

answer: the correct answer is -32

step-by-step explanation: