1. A Bates motor is a solid rocket motor with grains cast into multiple separate segments, each a cylindrical shape as shown. (Actually the sides taper in slightly, so the cross section is really trapezoidal instead of rectangular, but we will ignore this aspect here.) Wall -W. segment. (~ Side View End View (a) Find an expression for the burning surface area A of one motor segment, in terms of 11, 12 and w. Note that both ends burn as well as the inner bore of the cylinder. (b) As a motor segment burns, the burning surface recedes at a rate . Find an expression for the rate A at which the burning surface area changes, in terms of . (c) For a neutral burn (constant thrust), the burning surface area must be constant. What is the relation among 1, 2 and w such that A is zero? (d) Another desirable characteristic of a Bates motor is that the segments should burn to the wall at the same time as, or before, the width w goes to zero. What is the relation among 1, 2 and w such that the burn is neutral and r becomes equal to 12 (i.e., the segment burns to the wall) at the same time as w goes to zero?