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QUESTION

 A contractor rents out a piece of heavy equipment for $t$ hours
   and is paid \pounds 50 per hour. The equipment tends to overheat and
   if it overheats x times during the hiring period the contractor
   will have to pay a repair cost £$x^2$. The number of times the
   equipment overheats in $t$ hours can be assumed to have a poisson
   distribution with mean $2t.$ What value of $t$ will maximize the
   expected profit of the contractor?

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ANSWER

 If the equipment is hired for time $t$ and has $x$ faults then

   $P=50t-x^2$

   $E(P)=50t-E(X^2)$

   $X\sim P(2t)$ therefore

   \begin{eqnarray*}
    E(X)&=&2t\\
    \textrm{Var}(X)&=&E(X^2)-[E(X)]^2\\
    &=&2t-4t^2\\
    E(X^2)&=&2t+4t^2
   \end{eqnarray*}

   \begin{eqnarray*}
    E(P)&=&50t-2t-4t^2\\
    &=&48t-4t^2\\
    \frac{dE(P)}{dt}&=&48-8t=0 \textrm{ when }
    t=6\\
    \frac{d^2E(P)}{2t^2}&=&-8<0 \textrm{ hence maximum. }
   \end{eqnarray*}

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