Answer:
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 Â
y = [tex]\frac{40}{x^2}[/tex]
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 [tex](\frac{40}{x})[/tex]
f(x)  = 4x² +  [tex]\frac{160}{x}[/tex]
f'(x) Â = 8x - Â [tex]\frac{160}{x^2}=0[/tex]
f"(x) Â = [tex]\frac{8+320}{x^3}>0[/tex]
f'(x) Â = 0
[tex]\frac{8x-160}{x^2}=0[/tex]
x³ = 20
x = ∛20
x  = 2.71 cm
x²y = 40 Â
∛400y = 40
y = [tex]\frac{40}{\sqrt[3]{400} }[/tex]
 = [tex]\frac{40}{7.368}[/tex]
y  = 5.4288 ≈ 5.43 cm
Length and width = 2.71 cm and height = 5.43 cm
