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Problem 1. Consider the problem of finding the axial displacement of a truncated solid cone of length $L$ hanging under its own weight and subjected to a downward load $F$ at the tip as show in Fig. 1. The diameter at the top is $d_{0}$ and changes linearly to $d_{L}$ at the bottom. The BVP of this problem is as follows: $dxd?(EA(x)dxdu?)+?A(x)=0;0<x<L?A(X)=4??d(X)_{2};d(X)=d_{0}?Ld_{0}?d_{L}?Xu(0)=0;EAdxdu(L)?=F?$ (a) Derive a suitable weak form for use with the Galerkin method. Clearly indi cate how the boundary conditions will be handled. (b) Starting with a quadratic solution, obtain an approximate solution of the problem using MATLAB ( 5 extra points to develop the code) or Mathematica (similar codes are shared in Canvas). Submit the code and results. Consider $F=1000lb,L=100in,d_{0}=1in,d_{L}=41?in,?=1lb/in_{3}$, and $E=10_{6}lb_{in}in_{2}$.
Figure 1: Tapered hanging bar

(a) The weak form for this problem can be derived by multiplying the differential equation by a test function, v(x), and integrating over the domain:?