It is proposed to reduce the mass of an automobile by constructing the frame out of "aluminum alloys" rather than out of "low-carbon steel." The mass of a beam is proportional to the "materials performance index" of interest. We will simplify the analysis by assuming that: - the frame is a "beam" subject to bending loads only - the loads are independent of the material of the frame - the length and cross-sectional shape (but not the cross-sectional area) of the beam are fixed. a. Considering bending stiffness, what is the fractional weigh reduction, i.e., the mass of the aluminum alloy frame divided by the mass of the low-carbon steel frame? b. Considering bending strength, what is the fractional weight reduction? c. Since the frame must have sufficient stiffness and strength, what fractional weigh reduction is possible? (Hint: it will be your answer to a. or b.) d. Given your answer to c., what is the fractional change in the cost of materials? Neglect other costs such as shaping, joining and finishing. e. Now suppose that the most important objective is to increase the strength - to improve crash protection - rather than to reduce the mass. Your boss wants to minimize changes to the manufacturing process. Can you achieve the objective using some other form of steel?