let the length of the base is x and the height is y, then we have the width is x/2, the volume is x^2/2*y = 24
the cost c = 10*x^2/2 + 6*(2xy+2*x/2*y) = 5x^2 +18xy
y = 48/x^2
plug this into c
we have c = 5x^2+18x*48/x^2 = 5x^2+ 864/x
dc/dx = 10x - 864/x^2 = 0
x = 4.421
c = 5*4.421^2+864/4.421 = 97.719+195.431 = 293.15
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