The Lagrangian point L2 is very close to the most distant point from Earth with an umbra.
L2 is like the radius of the Hill sphere at r=asqrt[3]fracm3M for circular orbits, with m the mass of Earth, M the mass of the Sun, and a the distance Earth-Sun. The ratio fracm3M of the Earth and the triple mass of the Sun is almost exactly 10−6, the cubic root hence 0.01.
The diameter ratio of Earth and Sun is about 1/109. Therefore the umbra of Earth ends near 92 the distance to L2.
The answer to another bonus question would then be: If Earth would be 9 larger in diameter, but with the same mass, its umbra would end almost exactly at L2.
Earth's orbit isn't perfectly circular, but the aphel/perihel ratio of about 1.04 is insufficient to question the result qualitatively.
The error of the implicite assumptions tanx=x=sinx is negligible at the considered level of accuracy.
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