A rectangular box whose base is twice as long as it wide has volume of 256 cm^3. Material for the top costs 10 centavos per sq.m.; that for sides and bottom costs 5 centavos per sq. cm. Find the dimension that will make the coat to minimum, and find the cost.
x*2x*h = 256, so h = 128/x^2
So the cost
c(x) = 10*2x^2 + 5*2x^2 + 5*2(xh+2xh) = 30x^2 + 30x(128/x^2)
= 30x^2 + 384/x
dc/dx = 60x - 384/x^2 = (60x^3 - 384)/x^2
dc/dx=0 when x^3 = 384/60 = 32/5
so minimum cost is 144∛10