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# Parametric Equations

As a particle moves, its position can often be written in terms of time. Systems like these can be modeled with parametric equations, which cleverly write coordinates in terms of a parameter.

\[ \begin{cases} x = e^t \\ y = e^{2t} - 1 \end{cases} \]

What is the shape of the curve described by the above parametric equation?

Two clowns, Twinkle and Jingle, are throwing pies at each other. Twinkle throws a pie toward Jingle from \( 500 \) centimeters away. Its flight path is given by parametric equations \[ \begin{cases} x &=& 100t \\ y &=& 80t - 16t^2 \end{cases} \] where \(t\) is time in seconds.

Two seconds later Jingle launches an interceptor pie from his location with the flight path \[ \begin{cases} x &=& 500 - 500(t-2) \\ y &=& K(t-2) - 16(t-2)^2 \end{cases} \] Find the value of \(K\) which will guarantee that the interceptor pie will hit its target (the pie thrown by Twinkle).

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