Shadows on the ceiling

While dining at a restaurant, I notice a lamp descending from the ceiling, as shown in the diagram below. The lamp consists of a point light source at the center of a spherical bulb with a radius of 1 foot. The top half of the sphere is opaque. The bottom half of the sphere is semi-transparent, allowing light out (and thus illuminating my table) but not back in. The light source itself is halfway up to the ceiling—5 feet off the ground and 5 feet from the ceiling. The ground reflects light.

Above the light, on the ceiling, I see a circular shadow. What is the radius $R$ of this shadow?

Let's assume that the lightbulb is at the point $(0,5)$ and let's track the path of a ray of light that is emitted making an of $\theta$ with the $y$-axis as shown in the figure below. We see that it will bounce of the reflective floor at the point $(5\tan \theta, 0)$ and then return towards the ceiling with an angle of reflection of $\theta$ to make contact with the ceiling at the point $(15 \tan \theta, 10).$ The line from the bulb to the floor is given by $y = \left(-\cot \theta\right) x + 5,$ whereas the reflected line from the floor to the ceiling is given by $y= \left(\cot \theta\right) x - 5.$

To find the point along the reflected path from the floor to the ceiling we can either solve the optimization problem $$d^* = \min \{ (5+10\lambda)^2 \tan^2 \theta + (10 \lambda - 5)^2 \mid 0 \leq \lambda \leq 1 \}$$ either by some raw calculus or we can use the Lagrange multiplier theory and the knowledge that the line from the lightbulb to the point that minimizes the distance from the lightbulb should be perpendicular to the path of the light and some geometry. Going the geometric path, we see that since the hypotenuse of the right triangle has length $5 \sec \theta$ and the opposite angle measures $2\theta,$ so we have a distance of $$d^* = 5 \sec \theta \sin 2\theta = 10 \sec \theta \sin \theta \cos \theta = 10 \sin \theta.$$

Since the entire lamp, whether the opaque top or the semi-transparent lower half, would prevent the light from reaching the ceiling, we need to make sure that this minimal distance is at least 1, that is whenever $10 \sin \theta \geq 1$ then the ceiling will be illuminated. Therefore, the radius of the circular shadow is $$R = \min \{ 15 \tan \theta \mid 10 \sin \theta \geq 1 \} = 15 \tan \left( \sin^{-1} \frac{1}{10} \right) = \frac{15}{\sqrt{99}} = \frac{5}{\sqrt{11}} \approx 1.50755672289\dots,$$ since $\tan \left(\sin^{-1} x\right) = \frac{x}{\sqrt{1-x^2}}.$