A manufacturer wants to make a box with a hinged lid from a rectangular piece of

Question

A manufacturer wants to make a box with a hinged lid from a rectangular piece of cardboard that measures 17 by 12 inches. Six squares of sidelength x will be cut from the piece, one from each corner and one from the center of each long side, and then the ends and sides will be folded up (see figure). The volume of the box needs to be 72 cubic inches. The manufacturer asks you to find all possible values of x.

a) Write a polynomial whose zeros will give possible x values satisfying the above. Be sure to give your answer in standard form (FOIL it out).
b) List the possible rational roots of your polynomial according to the Rational Roots Theorem.
c) Use the three physical constraints of the problem— hint: the first one is x > 0, the other two come from the cardboard size— to define an interval where x must be in order for it to be a possible height of the box.
d) Now combine the previous 2 steps to make a shorter list of possible solutions x.
e) Next, test each of those x values until you find one possible solution (you may use Long Division or the Factor Theorem).
f) Use that one solution to find all others: That is, use the one solution you found and Long Division to find all other possible solutions x, and again, eliminate any that fail to satisfy the physical constraints.

Explanation / Answer

Length and width of sheet = 17x12

After cutting x on each side, dim of the box = (17-2x)(12-2x) x

a) Polynomial is

(17-2x)(12-2x) x = 72

4x^3-58x^2+204x-72 =0

The roots are = 0.396,4.974,9.13

c) x >0 Also 12-2x>0

Hence x <6

0<x<6

d) so x can be 0.396 or 4.974

e) When x =0.396 4x^3-58x^2+204x-72 =0

Also when x = 4.975 , f(x) =0

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