[1.18] Consider the following problem of launching a rocket to a fixed altitude b in a given time T while expending a minimum amount of fuel. Let u(t) be the acceleration force exerted at time t and let y(t) be the rocket altitude at time t. The problem can be formulated as follows: Minimizesubjectto0Tu(t)dty(t)=u(t)g 36 Chapter 1 y(T)=by(t)0,t[0,T], where g is the gravitational force and y is the second derivative of the altitude y. Discretize the problem and reformulate it as a linear programming problem. In particular, formulate the problem for T=15,b=20, and g=32. (Hint: Replace the integration by a proper summation and the differentiation by difference equations. Make the change of variables uj=xj,j, based on the discretization, and note that xjuj and xjuj.)