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555-605 (Medium)|   Geometry|      
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Bunuel
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AC is a semicircle and ABCD is a square. Alex and Brian each began at 21 Apr 2015, 06:09
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

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35% (medium)

Question Stats:

68% (02:02) correct 32% (02:02) wrong based on 162 sessions

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AC is a semicircle and ABCD is a square. Alex and Brian each began at point C and traveled to point A, with Alex taking the route of the semicircle to get there and Brian following the square from point C to point D, then from point D to point B, then from B to point A. If they each departed point C at the same time and then arrived at point A at the same time, what is the ratio of Alex's rate to Brian's rate?

A. π/6
B. π/4
C. π/3
D. π/2
E. 2π/3

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Semicircle_AC.png
Semicircle_AC.png [ 5.06 KiB | Viewed 4416 times ]
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A
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AC is a semicircle and ABCD is a square. Alex and Brian each began at 21 Apr 2015, 09:24
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Hi,
IMO it should be C pi/3
reason: alex travelled pi * r
brian travelled 3 times radius = 3r

ratio alex to brian (pi *r)/3r
pi/3

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Celestial


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Bunuel

AC is a semicircle and ABCD is a square. Alex and Brian each began at point C and traveled to point A, with Alex taking the route of the semicircle to get there and Brian following the square from point C to point D, then from point D to point B, then from B to point A. If they each departed point C at the same time and then arrived at point A at the same time, what is the ratio of Alex's rate to Brian's rate?

A. π/6
B. π/4
C. π/3
D. π/2
E. 2π/3

Kudos for a correct solution.

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Attachment:
Semicircle_AC.png
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AC is a semicircle and ABCD is a square. Alex and Brian each began at 22 Apr 2015, 00:46
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s be the side of the square. Thus radius of the semi-circle is s/2.
Now, Dist travelled by Alex = half the circumference of the semicircle = pi * (s/2)
Dist travelled by Brian = 3 sides of square = 3s
Since the time is same, (let A be the speed of Alex and B that of Brian)
=> pi*(s/2) / A = 3s / B
=> A/B = pi/6
Ans A.

Celestial09,
in your solution,
Quote:
brian travelled 3 times radius = 3r
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is wrong.
It should be
Quote:
brian travelled 3 times DIAMETER= 3r
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Dont you think?
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AC is a semicircle and ABCD is a square. Alex and Brian each began at 23 Apr 2015, 15:34
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Hi All,

In these types of situation, you can either approach the work with abstract Algebra and Geometry or you can TEST VALUES and use real values to solve some simple calculations.

IF...
The side of the square = 2 miles
The diameter of the circle = 2 miles
The radius of the circle = 1 mile

Alex travels the semi-circle = (1/2)(Circumference) = (1/2)(2)(pi)(1) = pi miles
Brian travels the 3 sides of the square = 3(2) = 6 miles

We're told that they both traveled for the same amount of TIME, so each of their rates are relative to the distance that they each traveled.

(pi miles)/(6 miles) = ratio of Alex's rate to ratio of Brian's rate = pi/6

Final Answer:
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A

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AC is a semicircle and ABCD is a square. Alex and Brian each began at 27 Apr 2015, 03:08
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Bunuel

AC is a semicircle and ABCD is a square. Alex and Brian each began at point C and traveled to point A, with Alex taking the route of the semicircle to get there and Brian following the square from point C to point D, then from point D to point B, then from B to point A. If they each departed point C at the same time and then arrived at point A at the same time, what is the ratio of Alex's rate to Brian's rate?

A. π/6
B. π/4
C. π/3
D. π/2
E. 2π/3

Kudos for a correct solution.

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VERITAS PREP OFFICIAL SOLUTION:

Since the diameter of the semicircle is one side of the square, you can start by assigning a variable (say, x) to the diameter. That means that Brian travels 3x as his distance, and for Alex you'll calculate half of the circumference of such a circle: 1/2*xπ. And since the times are the same, the ratio of the rates will be the same as the ratio of the distances. Alex traveled 1/2*xπ while Brian traveled 3x, so the x terms cancel and the ratio is π/2:3 which simplifies to π:6 making A the correct answer.
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AC is a semicircle and ABCD is a square. Alex and Brian each began at 21 Dec 2020, 18:32
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