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Question

# Solve the given problems.The vertical cross-section of a culvert under a road is elliptical. The culvert is $18$ m wide and $12$ m high. Find an equation to represent the perimeter of the culvert with the origin at road level and $2.0$ m above the top of the culvert.

Solution

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The center of the ellipse is at $(0,-8)$. Using the given dimensions for the ellipse, we can find the values of $a$ and $b$.

\begin{align*} \dfrac{\left(x-h\right)^2}{a^2}+\dfrac{\left(y-k\right)^2}{b^2}&=1\\ 2a&=18\longrightarrow a=9\\ 2b&=12\longrightarrow b=6 \end{align*}

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