2021 Fall AMC 10A Problem 17

Problem 17 of 25IntermediateAlgebra Geometry

An architect is building a structure that will place vertical pillars at the vertices of regular hexagon

ABCDEF,

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

A,

B,

and

C

are

12,

9,

and

10

meters, respectively. What is the height, in meters, of the pillar at

E?

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Solution

Put a regular hexagon in coordinates with

A=(-1,0),

B=(-\frac{1}{2},\frac{\sqrt3}{2}),

C=(\frac{1}{2},\frac{\sqrt3}{2}),

and

E=(\frac{1}{2},-\frac{\sqrt3}{2}).

Because the solar panel is flat, the height is an affine function

h(x,y)=ux+vy+w.

From

h(A)=12,

h(B)=9,

and

h(C)=10,

subtracting the last two equations gives

u=1.

Then

-u+w=12,

w=13.

Using

h(B)=9

gives

-\frac{1}{2}+\frac{\sqrt3}{2}v+13=9,

\sqrt3v=-7.

Therefore

\begin{aligned} h(E) &=\frac{1}{2}-\frac{\sqrt3}{2}v+13 \\ &=\frac{1}{2}+\frac{7}{2}+13 \\ &=17. \end{aligned}

Thus, D is the correct answer.

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