let the width of the box be x, then the length is 2x

let the height of the box be y

we want a box of the smallest surface area to minimize the cost.

we have V = lwh

=> V = x(2x)(y) = 2x^2 * y

we are told V = 10

=> 2x^2 * y = 10

=> y = 10/2x^2 = 5/x^2

now the surface area is given by

S = area of base + area of the sides

=> S = 2x^2 + 6xy

=> S = 2x^2 + 6x(5/x^2)

=> S = 2x^2 + 30x^-1

=> S' = 4x -30x^-2

for min surface area, set S' = 0

=> 4x - 30x^-2 = 0

=> 4x^3 - 30 = 0 .............multiplied through by x^2

=> 4x^3 = 30

=> x^3 = 30/4 = 15/2

=> x = cuberoot(15/2) ~= 1.9574

but y = 5/x^2 = 5/(1.9574)^2 ~= 1.305

so for min surface area: x = 1.9574, y = 1.305

so the area of the base = 2x^2 = 7.6628

so the cost of the base is $76.63

so the area of the sides is 6xy = 6(1.9574)(1.305) = 15.3264

so the cost of the sides is $91.96

so the cost of the material needed to make such a box is $91.96 + $76.63 = $168.59 ~= $169