If an open box is made from a tin sheet 6 in. square by cutting out identical squares from each corner and bending up the resulting flaps, determine the dimensions of the largest box that can be made. (Round your answers to two decimal places.)

4in. square x 4in. square x 1in. square

Step-by-step explanation:

By cutting out identical squares (considering 'x') from each corner and bending up the resulting flaps

The dimensions we have will be (see attachment for the figure)

V= (6-2x) (6-2x) x

As, Volume'V' = length (l) x width(w) x height(h)

V= (6x- 2x²)(6-2x)

V= 36x - 12x²-12x²+ 4x³

V=4x³ - 24x²+ 36x

Next is to find dV/dx,therefore we find the derivative and and set it to zero for the maximum volume

dV/dx = 12x² - 48x + 36

setting it to zero

12x² - 48x + 36 =0

x² - 4x + 3=0

x² -3x -x + 3=0

x(x-3) -1(x-3)=0

Either : x-3=0=> x=3

OR : x-1 =0 => x=1

Now, notice that 'x' cannot be 3 , because if  we cut 3 inch squares out of the original square, there  will be nothing left!

Also,  the volume will be 0 then. That  is the minimum volume, 0, when we cut all the tin away.

So, x=1

Therefore,

height 'x' = 1in. square

length and width = (6-2x) => 4in. square

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.

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step-by-step explanation: