CEMC Cayley, 2025

Geometry Difficulty 3.7 AMC 10/12 Find the answer Canada

In the diagram, line segment $CD$ has length $\text{48 cm}$ . Two ants, Violet and
Petunia, are at point $C$ and plan to
walk to point $D$ . Violet walks along
four congruent semi-circles above $CD$ . Petunia walks along three congruent
semi-circles below $CD$ .

Violet and Petunia start walking at the same time, and they walk at
constant, but different, speeds. Petunia arrives at $D$ $\text{12 seconds}$ before Violet and travels $3$ times as quickly as Violet. How long
does it take Violet to walk from $C$
to $D$ ?

Pick one

Solution

Violet walks along $4$
congruent semi-circles with diameters $CP=PQ=QR=RD=\dfrac{48\text{ cm}}{4}=12\text{ cm}$ .

Thus, the total distance that Violet walks is $4\times\dfrac12\pi(12\text{ cm})=24\pi\text{ cm}$ . Petunia walks along $3$
congruent semi-circles with diameters $CS=ST=TD=\dfrac{48\text{ cm}}{3}=16\text{ cm}$ .

Thus the total distance that Petunia walks is $3\times\dfrac12\pi(16\text{ cm})=24\pi\text{ cm}$ . Suppose that it takes Violet $t$ seconds to walk from $C$ to $D$ . Then it takes Petunia $t-12$ seconds to walk from $C$ to $D$ . Suppose that Violet travels at a
constant speed of $v$ cm/s. Then
Petunia travels at a constant speed of $3v$ cm/s. This information is summarized
in the table below.

	Distance (cm)	Time (s)	Speed (cm/s)
Violet	$24\pi$	$t$	$v$
Petunia	$24\pi$	$t-12$	$3v$

Since distance is equal to the product of speed and time, and each
ant travels the same distance, then $vt=3v(t-12)$ .

Since $\text{v gt;0}$ , then we can divide
both sides of the equation by $v$ ,
and solving for $t$ we get $t=3(t-12)$ or $t=3t-36$ or $36=2t$ , and so $t=18$ .

Thus, it takes Violet $18$ seconds to
walk from $C$ to $D$ .

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