A factory produces cartons each box has height h and base dimensions 2x, x and surface area A. Given that the capacity of a carton has to be 1030cm^3, (a) Using calculus find the value of x for which A is a minimum. (b) Calculate the minimum value of A.

To calculate the minimum value of A we first need to establish an equation for A. The surface area of a cuboid is relatively simple to figure out we simply work out the area of the faces and sum them together, this gives us the rather messy equation A = 4x² + 2xh + 4xh. We can simplify this by replacing one of the variables, in this instance h seems like a good candidate. By using the fact that the volume is 1030 cm^3, we can write, h = 515/ x², which after replacing h gives us, A = 4x² + 3090/x. Now we need to find the value of x for which A is a minimum. Since the question states we need to use calculus it is a good indication that this question will probably involve differentiation. We first want to find a turning or stationary point (i.e maxima or minima) which occur when the differential equation is equal to zero. So we begin by setting the differential to 0. f'(x) = 8x - 3090/ x², so at the mimimum (i.e when f'(x)=0), 8x - 3090/x² = 0, which means x_min=7.28 To double check it is a minimum we differentiate again and check that our x_min value when substituted in to the double differential is a positive value. Now to find the minimum value of A we simply substitute or x_min value in the equation for A which give our minimum value of A as 636 so A_min=636.