Answer:
715 cents
Explanation:
You want the minimum cost in cents for a rectangular box with a volume of 130 cm³ and a cost of top and bottom of 4 cents/cm² and a side cost of 5 cents/cm².
Short answer
The cost of each face is the same, so the height of the box will be 4/5 of the top or bottom edges. The dimensions of those will be the same as the edge dimension of a cube with volume (130 cm³)/(4/5) = ∛162.5 cm ≈ 5.457 cm.
The cost of the top is then (4 ¢/cm²)(5.457 cm)² ≈ 119.1 ¢. Each of the 6 faces has the same cost, so the total cost is ...
119.1¢ × 6 = 714.7¢ ≈ 715¢
The minimum cost of the box is about 715 cents.
Long Answer
We know the minimum cost of the top and bottom will be had when they are square. If we call the edge dimension of that square x, then its area is x², and the cost of the top and bottom together is ...
Ctb = 2 × (4¢/cm²)(x² cm²) = 8x² ¢
The height of the box can be found from the volume formula:
V = Bh
130 cm³ = (x² cm²)h
h = 130/x² cm
Then the side area is ...
side area = hx = (x cm)(130/x² cm) = 130/x cm²
And the cost of the four sides is ...
Cs = 4 × (5¢/cm²)(130/x cm²) = 2600/x ¢
The total cost is minimized when its derivative is zero.
C = 8x² +2600/x . . . . . . cents
dC/dx = 0 = 16x -2600/x²
Multiplying by x²/16, we have ...
x³ -2600/16 = 0
x = ∛162.5 ≈ 5.457 . . . . . cm
Then the cost is ...
C = 8(5.457)² +2600/5.457 = 238.23 +476.45 = 714.68 ≈ 715
The minimum cost of the box is 715 cents.
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Additional comment
The Longer Answer would be the optimization problem where you don't assume that the top and bottom are square. In solving that, you find out that they are, in fact, square.
A variation of this problem is to minimize the cost of an open-top box. In that case, the cost of the bottom is equal to the cost of two opposite sides.
Basically, every pair of opposite sides has the same total cost if the dimensions are unconstrained. If there is a specified aspect ratio, it reverts to a calculus problem.
The attached graph shows the total cost as a function of x, the bottom edge dimension.