Arectangular piece of cardboard can be turned into an open box by cutting away identical squares from each corner and folding up the resulting flaps. the cardboard is 12 inches long and 12 inches wide. a) write an expression for the length of the box in terms of x (the length of each square you cut away). b) write an expression for the width of the box in terms of x. c) write an expression for the volume of the box in terms of x. d) find the dimensions of the box that will yield the maximum volume.

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.

answer: just subtact the units and then add the non negative : happy to

step-by-step explanation: