2008 AMC 12A Problems/Problem 22

The following problem is from both the 2008 AMC 12A #22 and 2004 AMC 10A #25, so both problems redirect to this page.

Problem

A round table has radius $4$ . Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$ . Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$ ?

$[asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("$x$",(-1.55,2.1),E); label("$1$",(-0.5,3.8),S);[/asy]$

$\mathrm{(A)}\ 2\sqrt{5}-\sqrt{3}\qquad\mathrm{(B)}\ 3\qquad\mathrm{(C)}\ \frac{3\sqrt{7}-\sqrt{3}}{2}\qquad\mathrm{(D)}\ 2\sqrt{3}\qquad\mathrm{(E)}\ \frac{5+2\sqrt{3}}{2}$

Solution

Solution 1 (trigonometry)

Let one of the mats be $ABCD$ , and the center be $O$ as shown:

$[asy]unitsize(8mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("$x$",(-1.55,2.1),E); label("$x$",(0.03,1.5),E); label("$A$",(-3.6,2.5513),E); label("$B$",(-3.15,1.35),E); label("$C$",(0.05,3.20),E); label("$D$",(-0.75,4.15),E); label("$O$",(0.00,-0.10),E); label("$1$",(-0.1,3.8),S); label("$4$",(-0.4,2.2),S); draw((0,0)--(0,3.103)); draw((0,0)--(-2.687,1.5513)); draw((0,0)--(-0.5,3.9686));[/asy]$

Since there are $6$ mats, $\Delta BOC$ is equilateral. So, $BC=CO=x$ . Also, $\angle OCD = \angle OCB + \angle BCD = 60^\circ+90^\circ=150^\circ$ .

By the Law of Cosines: $4^2=1^2+x^2-2\cdot1\cdot x \cdot \cos(150^\circ) \Rightarrow x^2 + x\sqrt{3} - 15 = 0 \Rightarrow x = \frac{-\sqrt{3}\pm 3\sqrt{7}}{2}$ .

Since $x$ must be positive, $x = \frac{3\sqrt{7}-\sqrt{3}}{2} \Rightarrow C$ .

Solution 2 (without trigonometry)

See Math Jam

Since that link doesn't really lead to anything now, here it is:

Draw the center of the circle. Then connect the radius to a far corner of the box and a partial radius to the closer corner of the box.

Then, drop a perpendicular down between the radiis.

Now we write the length of perpendiculars in terms of the segmented radii and the full radii.

$[asy]unitsize(8mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("$x$",(-1.55,2.1),E); label("$x$",(0.03,1.5),E); label("$A$",(-3.6,2.5513),E); label("$B$",(-3.15,1.35),E); label("$C$",(0.05,3.20),E); label("$D$",(-0.75,4.15),E); label("$O$",(0.00,-0.10),E); label("$1$",(-0.1,3.8),S); label("$4$",(-0.4,2.2),S); draw((0,0)--(0,3.103)); draw((0,0)--(-2.687,1.5513)); draw((0,0)--(-0.5,3.9686));[/asy]$

Since i can't write latex stuff, lets skip to the end. (lolol)

we have x^2 + xsqrt3 - 15 = 0. Utilize quadratic equation.

We now have x = (-3 + sqrt63)/2

take out the 9 from the sqrt and we have:

(3sqrt7 - 3)/2, answer choice C.

2008 AMC 12A (Problems • Answer Key • Resources)
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All AMC 12 Problems and Solutions

2008 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Last Question
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2008 AMC 12A Problems/Problem 22

Contents

Problem

Solution

Solution 1 (trigonometry)

Solution 2 (without trigonometry)

See Also