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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 $D$ $12 $\text{}$
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}$.

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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 $v>0$ , 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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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.