A company must manufacture a closed rectangular box with a square base. The volume must be 1120 cubic inches.?

Question

The top and the bottom squares are made of a material that costs 7 dollars per square inch. The vertical sides are made of a different material that costs 6 dollars per sqare inch.
What is the minimal cost of a box of this type?

Answer 1

Minimal cost of the box is $4087.240 (approx)

Let me try to attempt this.

Let l, w and h correspond to the length, breadth and height of the rectangular box in inches, and let x be the least cost in dollars.

Then lwh = 1120,

And cost of materials,
x = 7 * (2lw) + 6 * 2 * (h(l+w)), or
x = 14lw + 12h*(l+w)

First of all, let's prove that l = w.
Well, I shall leave it to you because I am too lazy, but a hint is to prove by contradiction (suppose l > w, then we cannot have the least cost).

So since l = w, we have

hw^2 = 1120 or h = 1120 / (w^2), and
x = 14w^2 + 24hw

Substitute the first eqn in:
x = 14w^2 + 26880/w

Differentiate that:
dx/dw = 28w - 26880/ (w^2) = 0
Clearly w^2 is not zero so we can multiply that in:
28w^3 - 26880 = 0
w^3 = 960

Checking the real value of w with the second derivation we get:
d2x/dx2 = 28 + 53760/960 = 84 > 0 proving that it is indeed the minimum value.


Hence, the minimal cost of the box, x
= 14w^2 + 26880/w
= 14 * 960^(2/3) + 26880/ 960^(1/3)
= 4087.240 (approx)


Or we can also find h then substitute back into the original equation. Same thing.

Edit: It looks like Hiker has got a nose faster than me in submitting his answer haha.

2nd edit: Hiker, your answer, $4043.45, looks wrong, because your h = 1120 / 9.86485^2 = 11.32402 is wrong. h = 11.50899! If you substitute that in, you will get the same answer as my answer, and also your approximation: $4088 (approx)

Answer 2

Top and bottom are squares, let w = the side of 1 square, in inches.
Let h = the height of the box.
Each side is w x h square inches.
Top and bottom are each w x w = w^2 square inches.
Cost = 7 (Top area + bottom area) + 6 x 4 x (Area of one side)
= 7 (w^2 + w^2) + 24 (wh)
= 14w^2 + 24wh
Volume = w^2 h = 1120
h = 1120 / (w^2)
Cost = 14w^2 + 24wh
= 14 w^2 + 24 w (1120/w^2)
= 14w^2 + 26880 w^(-1)
dCost/dw = 28w -26880 w^(-2)
Let dCost/dw = 0
28w -26880 w^(-2) = 0
28w - 26880/ (w^2) = 0
{Multiply both sides by w^2}

28w^3 - 26880 = 0
28w^3 = 26880
w^3 = 26880
w= 26880^(1/3) = 9.86485 in.
h = 1120 / (w^2)
h = 1120 / 9.86485^2 = 11.32402 in.
Cost = 14w^2 + 24wh
= 14 (9.86485^2) + 24 (9.86485)(11.32402)
= $4043.45
It sounds like a lot, but we need 1120 cubic inches of volume.
sides are approximately 112 square in., which would come in at $672 per side. Top is about 100 square in., which comes to $700 for top, and $700 per bottom.
Add up: 4 sides + 1 top + 1 bottom
= 4 * 672 + 700 + 700 = 4088 (This is an approximation, but yes, it's over 4,000 dollars per box. The answer, $4043.45, looks right.)