Ajewelry box with a square base is to be built with copper plated sides, nickel plated bottom and top, and a volume of 40cm^3. if nickel plating costs $2 per cm^2 and copper plating costs $1 per cm^2, find the dimensions of the box to minimize the cost of the materials. write out the process of solving this question in plain english

Length and width = 2.71 cm and height = 5.43 cm

Step-by-step explanation:

Let the length of the jewelry box = x

Let the width = x          [box is square, so length and width are same]

Let the height = y

Given volume = 40 cm³

Therefore, V = x²y

x²y = 40  

y =

Cost of Nickle plating on top and bottom = $2 per cm²

Cost of copper plating on sides = $1 per cm²

Cost = 2(Areas of top and bottom) + 1(Areas of sides)

       = 2(x² + x²) + 1(xy + xy + xy + xy)

       = 4x² + 4xy

       = 4x² = 4x

f(x)   = 4x² +  

f'(x)  = 8x -  

f"(x)  =

f'(x)  = 0

x³ = 20

x = ∛20

x  = 2.71 cm

x²y = 40  

∛400y = 40

y =

  =

y  = 5.4288 ≈ 5.43 cm

Length and width = 2.71 cm and height = 5.43 cm

in problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion. to find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

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