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\title{\Huge{Calculus III}\\Homework \# 1}
\author{\huge{Krishna Ayyalasomayajula}}
\date{}

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\chapter{Lab 2 - 13.1 Apps}
\section{Work}

\qs{}{
  Let $\vec{T_1}$ represent the tension of the leftmost cable, while $\vec{T_2}$ encodes the tension force experienced by the rightmost cable. Our coordinate system will originate at the intersection of the two cables.

  \begin{align*}
    \vec{T_1} \coloneqq \langle \|\vec{T_1}\| ; 135\degree \rangle \\
    \vec{T_2} \coloneqq \langle \|\vec{T_2}|\| ; 15\degree \rangle \\
    500=\|\vec{T_1}\|\sin{135\degree} + \|\vec{T_2}\|\sin{15} \\
    0 = \|\vec{T_1}\|\cos{135\degree} + \|\vec{T_2}\|\cos{15} \\
    \implies 500=\|\vec{T_1}\|\tfrac{\sqrt{2}}{2} + \|\vec{T_2}\|\cdot0.258819045103 \\
    \implies0 = \|\vec{T_1}\|\tfrac{-\sqrt{2}}{2} + \|\vec{T_2}\|\cdot0.965925826289 \\
    \text{Solving the system numerically yields: }\\
    \|\vec{T_1}\| \Rightarrow 557.677\, \mathrm{lb}\\
    \|\vec{T_2}\| \Rightarrow 408.248\,\mathrm{lb} \\
    \text{This corresponds to answer choice A}
  \end{align*}
  
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