A searchlight is shaped like a paraboloid of revolution, where its light source is located 2ft from base along the axis of symmetry and its opening is 5ft across. How deep should searchlight be?
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To solve this problem, you must first figure out the best way to attack the problem.
Parabolas have many forms in terms of their equations, and each form is useful in different situations. Considering that the light source in a parabolic light will be situated at the focus (because the reflected light ray will go outward in a direction parallel to the axis of rotation), we should use a parabola equation that involves the focus. I would suggest the following form:
`y = 1/(4p) (x-h)^2 + k`
Here, `(h,k)` is the vertex or base of the parabola and `p` is the distance from the vertex to the focus. We are given this distance, so we can already set `p = 2`:
`y = 1/8(x-h)^2 + k`
Considering that we haven't been given a coordinate system to work with, we can define one however we please! Let's make this simple and define it so that our vertex is at the origin, giving us our final parabola equation:
`y = 1/8x^2`
Here is a graph to better visualize:
Now to calculate the depth, we need to use the second parameter given in the problem: that the parabola is 5 feet across. Considering our coordinate system, this means that we are looking for the `y` at the `x`-values `+-2.5`, seen here:
If you are having trouble with this concept, convince yourself that the the line segment from `(-2.5, y)` to `(2.5,y)` is horizontal and has a length of 5 using the distance and slope formulas.
Now, we simply need to solve for the `y`, which will give us the total depth of the reflective dish:
`y = 1/8(2.5)^2`
`y = 1/8(6.25) ~~ 0.78`
Therefore, our reflective dish has a depth of 0.78 ft.
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