Answer:
Base= 5.09 cm x 5.09 cm; height = 1.69 cm
Step-by-step explanation:
-> materials has a square base of side length, dimension will be: x . x = x²
'y' represents height
->For dimensions of 4 silver plated sides= xy each
->dimensions of the nickel plated top= x²
Volume = yx²
44=yx² => y= 44/x²
Cost of the sides will be( 4 * xy * $3 )
Cost of the top and the bottom will be (2 * x² * $1)
For the Total cost: 12xy + 2x²
substituting value of 'y' in above equation,
=> Total cost = 12x (44/x²) + 2x² = 528 / x + 2x²
To Minimum critical point => d [cost] / dx = 0
=> - 528/x² + 4x =0
132/x² - x =0
132 - x³ = 0
x³ = 132
Taking cube root on both sides
∛x³ = ∛(132)
x= 5.09
=> y = 44/5.09² =>1.69
Dimensions of the box :
Base= 5.09 cm x 5.09 cm; height = 1.69 cm
