Question

A jewelry box with a square base is to be built with silver plated sides, nickel plated bottom and top, and a volume of 44 cm3. If nickel plating costs $1 per cm2 and silver plating costs $3 per cm2, find the dimensions of the box to minimize the cost of the materials. (Round your answers to two decimal places.) The box which minimizes the cost of materials has a square base of side length

Answer 1

Base= 5.09 cm x 5.09 cm; height = 1.69 cm

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