Question

Minimizing Packaging Costs

A rectangular box is to have a square base and a volume of 54 ft³. If the material for the base costs $0.22/ft², the material for the sides costs $0.09/ft², and the material for the top costs $0.14/ft², determine the dimensions (in ft) of the box that can be constructed at minimum cost. (Refer to the figure below.)

A closed rectangular box has a length of x, a width of x, and a height of y.

x=? ft

y= ? ft

Answer 1

and volume 54 ft 3 x2 y = 54 - xy = 5488 Simce the material box have square bon base Then let area for base es x² ft2 that is it has sedes at se ft y ft. and heigh by ; and the costs of base, top and sides we can write the total cast as : C(x,y) = $ ( 3 x 0.22 + 4xy xo•og + x ² x 0.14) (A C (W) = $ ( X² XO:22 + 4x54 x 0.09 + 2X0.14) ". The cost function becomes CCW) ge² x (0*36) + 18:44 re too Now, c'(x) = 0.72 se 19.44 Dez for c'(n) 19.44 Or 72 x 0441 3 19.44 - - 27 0.72 A)

0.72 t 19.44 x 2 x 3 - Cus) > 0 C (x) is minimum . Far; n = 3 ; a 54 6 y = x2 A closed rectangular bax has a length afx, a width af N. and a height of y Then : R = 3 ft ft 6 كه