I need help with question 10 please

172

Step-by-step explanation:

13/40=x/528

40x=6864

x=172

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.