Answer:
4.5 inches
Step-by-step explanation:
To find the depth of the parabolic dish, let's consider the equation of a vertical parabola with its vertex at the origin (0, 0).
The general equation for a parabola with a vertical axis of symmetry is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Standard form of a vertical parabola}}\\\\(x-h)^2=4p(y-k)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$p\neq 0$}\\\phantom{ww}\bullet\;\textsf{Vertex:\;$(h,k)$}\\\phantom{ww}\bullet\;\textsf{Focus:\;$(h,k + p)$}\\\phantom{ww}\bullet\;\textsf{Directrix:\;$y=(k-p)$}\\\phantom{ww}\bullet\;\textsf{Axis of symmetry:\;$x=h$}\end{array}}[/tex]
In this case, the vertex is the origin (0, 0), so h = 0 and k = 0.
The focus is always on the inside of the parabola.
Since p represents the distance from the vertex to the focus, and the distance from the vertex to the focus is given as 8 inches, then p = 8.
Therefore:
[tex]\begin{aligned}(x-0)^2&=4(8)(y-0)\\\\x^2&=32y\\\\y&=\dfrac{1}{32}x^2\end{aligned}[/tex]
The diameter of the dish is 24 inches, indicating that the width from one side of the parabola to the other is 24 inches. The width is determined by the distance between two symmetric points on the parabola. Since the axis of symmetry is the y-axis, the two symmetric points are (12, y) and (-12, y). The y-value of these points represents the depth of the parabolic dish when its diameter is 24 inches. Therefore, to find the depth, substitute x = 12 into the equation of the parabola:
[tex]\begin{aligned}y&=\dfrac{1}{32}(12)^2\\\\y&=\dfrac{1}{32}(144)\\\\y&=\dfrac{144}{32}\\\\y&=4.5\; \sf inches\end{aligned}[/tex]
Therefore, the depth of the parabolic dish is 4.5 inches.
