A transport container for radioactive waste is designed as a cuboid with width xxx m, length 2.5x2.5x2.5x m, and height hhh m. The total interior volume of the container must be 500 m3500\text{ m}^3500 m3.
Show that the total surface area of the container, S m2S\text{ m}^2S m2, is given by
S=5x2+1400x S = 5x^2 + \frac{1400}{x} S=5x2+x1400Find dSdx\frac{\text{d}S}{\text{d}x}dxdS.
Hence find the value of xxx for which SSS is stationary, giving your answer to 3 significant figures.
Find d2Sdx2\frac{\text{d}^2S}{\text{d}x^2}dx2d2S and hence verify that the value of xxx found in part (c) gives a minimum value for SSS.
Calculate the minimum surface area of the container, giving your answer to 1 decimal place.
