A material is required for a cheap column with a solid circular cross section that must support a load F without contracting along its length L more than d. Write down an equation for the material cost of the column in terms of its dimensions, the price per kg of the material, Cm, and the material density r. The cross-section area A is a free variable; eliminate it by using the constraint on d. Hence derive the material index for finding the cheapest-stiffest column. Plot the index on a copy of the appropriate chart (from CES EduPack or copied from the textbook), and identify the best three candidate materials. The chart must be submitted with your homework to illustrate your answer.