College Algebra, Chapter 8, 8.1, Section 8.1, Problem 56

Suppose that a telescope has $200$-in mirror that is constructed in a parabolic shape that collects light from the stars and focuses it at the prime focus, that is, the focus of the parabola. The mirror is $3.79$-in deep at its center. Determine the distance from the vertex to the focus.



If we let the vertex of the parabola lies on the origin and halfway between $200$-in. Then its equation is $x^2 = 4py$ where the focus is located at $(0, p)$ and endpoints at $(100, 3.79)$ and $(-100, 3.79)$. Hence, the endpoints are the solution of the equation, so..


$
\begin{equation}
\begin{aligned}

x^2 =& 4py
\\
\\
(100)^2 =& 4p(3.79)
\\
\\
p =& 659.6306 \text{-in}

\end{aligned}
\end{equation}
$


It shows that the distance from the vertex to the focus is approximately $660$ inches.

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