Lagrange Multipliers

Due by Mon, 19 Mar 2018

Consider two rectangles with length $L$ and breadth $B$ placed one on top of the other to form a $L \times 2B$ rectangle. Consider a connected path joining the bottom left corner of the lower rectangle to the top right corner of the upper rectangle consisting of two line segments $l_1$ and $l_2$, one in each rectangle and meeting at the common edge.

  1. Let $\theta_1$ and $\theta_2$ be the angles made by $l_1$ and $l_2$ to the line perpendicular to the common edge. Then show that $B\cdot(\tan(\theta_1) + \tan(\theta_2)) = L$.
  2. If light travels along $l_1$ and $l_2$ at speeds $v_1$ and $v_2$, respectively, find the total time travelled $T(\theta_1, \theta_2)$ as a function of $\theta_1$ and $\theta_2$.
  3. Show that the path (of the above form) along which the total time travelled is minimum satisfies Snell’s law $\sin(\theta_1)/\sin(\theta_2) = v_1/v_2$.
  4. Consider all cuboids with length $L$ breadth $B$ and height $H$ with surface area $6$. Show that the cuboid with largest volume among these is the cube with unit side.

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