Difference between revisions of "2021 Fall AMC 10A Problems/Problem 17"
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+ | ==Solution 2== | ||
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+ | Let the height of the pillar at <math>D</math> be <math>x.</math> Notice that the difference between the heights of pillar <math>C</math> and pillar <math>D</math> is equal to the difference between the heights of pillar <math>A</math> and pillar <math>F.</math> So, the height at <math>F</math> is <math>x+2.</math> Now, doing the same thing for pillar <math>E</math> we get the height is <math>x+3.</math> Therefore, we can see the difference between the heights at pillar <math>C</math> and pillar <math>D</math> is half the difference between the heights at <math>B</math> and <math>E,</math> so | ||
+ | <cmath>x+3-9=2 \cdot (x-10) \implies x-6=2 \cdot (x-10) \implies x=14 \implies x+3=\boxed{17}.</cmath> | ||
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+ | - kante314 |
Revision as of 21:06, 22 November 2021
Problem
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon , which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at
,
, and
are
,
, and
meters, respectively. What is the height, in meters, of the pillar at
?
Solution
Since the pillar at has height
and the pillar at
has height
and the solar panel is flat, the inclination from pillar
to pillar
would be
. Call the center of the hexagon
. Since
is parallel to
,
has a height of
. Since the solar panel is flat,
should be a straight line and therefore, E has a height of
=
.
~Arcticturn
Solution 2
Let the height of the pillar at be
Notice that the difference between the heights of pillar
and pillar
is equal to the difference between the heights of pillar
and pillar
So, the height at
is
Now, doing the same thing for pillar
we get the height is
Therefore, we can see the difference between the heights at pillar
and pillar
is half the difference between the heights at
and
so
- kante314