Question

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 $E?$

$9$

$6\sqrt{3}$

$8\sqrt{3}$

$17$

$12\sqrt{3}$

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 $-u+w=12$ , so $w=13$ . Using $h(B)=9$ gives $-\frac{1}{2}+\frac{\sqrt3}{2}v+13=9$ , so $\sqrt3v=-7$ .

Therefore $h(E)=\frac{1}{2}-\frac{\sqrt3}{2}v+13=\frac{1}{2}+\frac{7}{2}+13=17.$

Thus, D is the correct answer.

2021 AMC 10A Fall Problem 17

Solution:

Problem 17 in Other Years