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2021 JMC 10 Problems/Problem 17

Problem

One lit lightbulb is $2$ units above the top of spherical ball with a radius of $8.$ The spherical ball, lying atop a flat floor, casts a shadow. What is the area of this shadow?

$\textbf{(A) } 100\pi \qquad\textbf{(B) } 144\pi \qquad\textbf{(C) } 288\pi \qquad\textbf{(D) } 576\pi \qquad\textbf{(E) } 1024\pi$


Solution

Consider the cross section of the sphere below. The direction of the light is indicated by the dashed lines and the shadow is indicated by the purple segment. The center of the sphere, one of the tangent points, and the lightbulb form a triangle with sides $6,8,$ and $8+2=10.$ This is a right triangle, similar to one of the larger right triangles in the cross section with legs of length $r$ and $8+8+2=18.$ It follows that $\tfrac{8}{6} = \tfrac{r}{18} \implies r = 24.$ So the shadow is a circle with radius $24,$ and area is $24^2 \cdot \pi = 576\pi.$

[asy] size(4.5cm); draw(circle((0,8),8)); pair A = (0,0); pair B=(0,8); pair C=(0,18); pair D = (-4.8, 14.4); pair E = (4.8, 14.4); pair F=(-24,0); pair G=(24,0); draw(F--C--G, dashed); draw(F--G, purple); draw(A--C); draw(D--B--E); label("8",(-2.4, 11.2), SW); label("8",(2.4, 11.2), SE); label("6",(-2.4, 16.2), NW); label("6",(2.4, 16.2), NE); label("$r$",(-12, 0), S); label("$r$",(12, 0), S); [/asy]

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