The Height of Lights

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In parsha Beha’aloscha (Bamidbar 8:2) the Cohen is commaned to light the menorah in the temple.  Ramban explains that our lighting of the Chanaukah menorah is a continuation of this mitzvah.

 

Author: S. B. Schramm
Math Level: Geometry
Grade Level: 10th-11th
Topics: Chanukah candles

The Question:
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Yehudit is checking to make sure that her candles are between the recommended 3 and 10 tfachim above the ground.  Standing 10 amot away she looks down at her candles at an angle of 9°.  If her eye height is 3 amot, are her candles at the correct height? (Note: 6 tfachim is equivalent to 1 amah)

According to the Chazon Ish 1 tefach is equivalent to 9.6 cm.  What is the height of Yehudit’s candles?

According The Chatam Sofer, 1 tefach is approximately 4.25 inches.  What is the height of Yehudit’s candles?

Reuven is walking along when he spots a chanukia in the second story window and wonders if it is below the maximum height of 20 amot above the ground.  When he is 50 amot away from the building he looks up to the candles at an 18⁰ angle and down to the base of the building at a 3⁰ angle.  How high are the candles?  Are they below the maximum?

The Answer:
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Let h be the height of the chanukiah and x be the distance from Yehudit’s eyes down to the candles.
x/10=tan9°   so x=10 ∙tan9°=1.58 amot
h=eye height-x=3-1.58=1.42 amot
(6 tfachim)/(1 amah)= h/(1.42 amot)
h=6 ∙1.42=8.52 tfachim
Her chanukiah is at good height.

(9.6 cm)/(1 tefach)= h/(8.52 tfachim)
h=9.6 ∙8.52=81.79 cm

(4.25 inches)/(1 tefach)= h/(8.52 tfachim)
h=4.25∙8.52=36.21 inches

Let a be the distance from Reuven’s eyes up to the candles and b be the distance to the base of the building.  Thus
h=a+b
a/50=tan18°    so a=50∙tan18°=16.24 amot
b/50=tan3°    so b=50∙tan3°=2.62 amot
height of chanukiah is 16.24+2.62=18.86 amot
The chanukiah is below the maximum.

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