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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 $D$ $\text{12 seconds}$ 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}. [[IMAGE0]] Thus, the total distance that Violet walks is

Figure for this problem $4\times\dfrac12\pi(12\text{ cm})=24\pi\text{}$
cm} $. Petunia walks along$ 3 congruent semi-circles with diameters

Figure for this problem $CS=ST=TD=\dfrac{48\text{ cm}}{3}=16\text{}$
cm}. Thus the total distance that Petunia walks is

Figure for this problem $3\times\dfrac12\pi(16\text{ cm})=24\pi\text{}$
cm}. Suppose that it takes Violet

Figure for this problem 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

Figure for this problem v cm/s. Then Petunia travels at a constant speed of

Figure for this problem 3v cm/s. This information is summarized in the table below. Distance (cm) Time (s) Speed (cm/s) Violet

Figure for this problem $24\pi$ tv $Petunia$ $24\pi$ t-123v Since distance is equal to the product of speed and time, and each ant travels the same distance, then

Figure for this problem vt=3v(t-12) $. Since$ v>0, then we can divide both sides of the equation by

Figure for this problem 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.