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Manager  Joined: 18 Oct 2010
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A cylinder has a base with a circumference of 20pi meters  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 71% (02:34) correct 29% (02:46) wrong based on 386 sessions

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A cylinder has a base with a circumference of 20pi meters and an equilateral triangle inscribed on the interior side of the base. A marker is dropped into the tank with an equal probability of landing on any point on the base. If the probability of the marker landing inside the triangle is (Sqrt 3)/4 , what is the length of a side of the triangle?

A. 3(sqrt 2pi)
B. 3(sqrt 3pi)
C. 10*sqrt (pi)
D. 10(sqrt 3pi)
E. 20 pi

Originally posted by Joy111 on 07 May 2012, 03:12.
Last edited by Joy111 on 07 May 2012, 05:06, edited 1 time in total.
Math Expert V
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Posts: 59587
Re: A cylinder has a base with a circumference of 20pi meters  [#permalink]

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4
Joy111 wrote:
A cylinder has a base with a circumference of 20pi meters and an equilateral triangle inscribed on the interior side of the base. A marker is dropped into the tank with an equal probability of landing on any point on the base. If the probability of the marker landing inside the triangle is (Sqrt 3)/4 , what is the length of a side of the triangle?

A. 3(sqrt 2pi)
B. 3(sqrt 3pi)
C. 10 pi
D. 10(sqrt 3pi)
E. 20 pi

Since the probability of the marker landing on the portion of the base inside the triangle is $$\frac{\sqrt{3}}{4}$$ then the portion of the base (circle) inside the triangle must be $$\frac{\sqrt{3}}{4}$$ of the area of the base.

Next: $$circumference=20\pi=2\pi{r}$$ --> $$r=10$$ --> $$area_{base}=\pi{r^2}=100\pi$$;

The area of the equilateral triangle is $$\frac{\sqrt{3}}{4}$$ of the base: $$area_{equilateral}=\frac{\sqrt{3}}{4}*100\pi$$ --> also the ares of the equilateral triangle is $$area_{equilateral}=a^2*\frac{\sqrt{3}}{4}$$, where $$a$$ is the length of a side --> $$area_{equilateral}=a^2*\frac{\sqrt{3}}{4}=\frac{\sqrt{3}}{4}*100\pi$$ --> $$a=10{\sqrt\pi}$$.

Similar question to practice: a-cylindrical-tank-has-a-base-with-a-circumference-of-105453.html

Hope it helps.
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Joined: 24 Jul 2011
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GMAT 1: 780 Q51 V48 GRE 1: Q800 V740 ### Show Tags

4
Prob of marker landing in the triangle = area of triangle / area of base = sqrt(3)/4

Circumference of base = 20pi meters
=> Radius of base = 10 meters
=> Area of base = 100pi sq m

Therefore area of the triangle = 100pi * sqrt(3)/4

As the area of an equilateral triangle is sqrt(3)*(side^2)/4,
any side of the triangle = 10*sqrt(pi)

This doesn't seem to be any of the answer choices.
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Manager  Joined: 18 Oct 2010
Posts: 71

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GyanOne wrote:
Prob of marker landing in the triangle = area of triangle / area of base = sqrt(3)/4

Circumference of base = 20pi meters
=> Radius of base = 10 meters
=> Area of base = 100pi sq m

Therefore area of the triangle = 100pi * sqrt(3)/4

As the area of an equilateral triangle is sqrt(3)*(side^2)/4,
any side of the triangle = 10*sqrt(pi)

This doesn't seem to be any of the answer choices.

sorry missed the sqrt in option C

It has now been edited , My bad A. 3(sqrt 2pi)
B. 3(sqrt 3pi)
C. 10 sqrt (pi)
D. 10(sqrt 3pi)
E. 20 pi
Intern  Joined: 08 Feb 2011
Posts: 7
Re: A cylinder has a base with a circumference of 20pi meters  [#permalink]

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3
Bunuel wrote:
Joy111 wrote:
A cylinder has a base with a circumference of 20pi meters and an equilateral triangle inscribed on the interior side of the base. A marker is dropped into the tank with an equal probability of landing on any point on the base. If the probability of the marker landing inside the triangle is (Sqrt 3)/4 , what is the length of a side of the triangle?

A. 3(sqrt 2pi)
B. 3(sqrt 3pi)
C. 10 pi
D. 10(sqrt 3pi)
E. 20 pi

Since the probability of the marker landing on the portion of the base inside the triangle is $$\frac{\sqrt{3}}{4}$$ then the portion of the base (circle) inside the triangle must be $$\frac{\sqrt{3}}{4}$$ of the area of the base.

Next: $$circumference=20\pi=2\pi{r}$$ --> $$r=10$$ --> $$area_{base}=\pi{r^2}=100\pi$$;

The area of the equilateral triangle is $$\frac{\sqrt{3}}{4}$$ of the base: $$area_{equilateral}=\frac{\sqrt{3}}{4}*100\pi$$ --> also the ares of the equilateral triangle is $$area_{equilateral}=a^2*\frac{\sqrt{3}}{4}$$, where $$a$$ is the length of a side --> $$area_{equilateral}=a^2*\frac{\sqrt{3}}{4}=\frac{\sqrt{3}}{4}*100\pi$$ --> $$a=10{\sqrt\pi}$$.

Similar question to practice: a-cylindrical-tank-has-a-base-with-a-circumference-of-105453.html

Hope it helps.

Hello All,

See my solution below and please tell me where I am going wrong:
Radius of base = 10 (as derived by you)
Now, if we draw the equilateral triangle inscribed in a circle (as shown in my attachment which is not to scale), then:
and O is the centroid
Centroid divides a median in the ratio 2:1. Hence, OD = OA / 2 = 5

Now, AD is the height of the triangle and in equilateral triangle,
[math]height = a * sqrt (3) /2
where a = side of triangle

Hence, 15 = a * sqrt (3) /2
Hence a = 10 * sqrt (3)

Where am I going wrong?
Attachments Circle.jpg [ 16.8 KiB | Viewed 9157 times ]

Intern  Joined: 18 Jun 2012
Posts: 31
Re: A cylinder has a base with a circumference of 20pi meters  [#permalink]

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In this question , Triangle is inscribed inside circle so if I am not wrong, centre of the circle will be the circumcentre. As per Maths book of GMAT club relationship between Equilateral triangle and Circcumradius is
R = a/3^1/2 . We know R = 10 , so why putting the values in this formula getting the right answer ?
Intern  Joined: 03 Oct 2010
Posts: 4
Re: A cylinder has a base with a circumference of 20pi meters  [#permalink]

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In this question , Triangle is inscribed inside circle so if I am not wrong, centre of the circle will be the circumcentre. As per Maths book of GMAT club relationship between Equilateral triangle and Circcumradius is
R = a/3^1/2 . We know R = 10 , so why putting the values in this formula NOT getting the right answer ?
Manager  Joined: 12 Feb 2012
Posts: 114
Re: A cylinder has a base with a circumference of 20pi meters  [#permalink]

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Bunuel wrote:
Joy111 wrote:
A cylinder has a base with a circumference of 20pi meters and an equilateral triangle inscribed on the interior side of the base. A marker is dropped into the tank with an equal probability of landing on any point on the base. If the probability of the marker landing inside the triangle is (Sqrt 3)/4 , what is the length of a side of the triangle?

A. 3(sqrt 2pi)
B. 3(sqrt 3pi)
C. 10 pi
D. 10(sqrt 3pi)
E. 20 pi

Since the probability of the marker landing on the portion of the base inside the triangle is $$\frac{\sqrt{3}}{4}$$ then the portion of the base (circle) inside the triangle must be $$\frac{\sqrt{3}}{4}$$ of the area of the base.

Next: $$circumference=20\pi=2\pi{r}$$ --> $$r=10$$ --> $$area_{base}=\pi{r^2}=100\pi$$;

The area of the equilateral triangle is $$\frac{\sqrt{3}}{4}$$ of the base: $$area_{equilateral}=\frac{\sqrt{3}}{4}*100\pi$$ --> also the ares of the equilateral triangle is $$area_{equilateral}=a^2*\frac{\sqrt{3}}{4}$$, where $$a$$ is the length of a side --> $$area_{equilateral}=a^2*\frac{\sqrt{3}}{4}=\frac{\sqrt{3}}{4}*100\pi$$ --> $$a=10{\sqrt\pi}$$.

Similar question to practice: a-cylindrical-tank-has-a-base-with-a-circumference-of-105453.html

Hope it helps.

Hey Bunuel quick question,

If the radius of a circle that inscribed an equaliteral is $$r=S\sqrt{3}/3$$, where r is the radius and S is the side of the equilateral, should int the answer be $$30/\sqrt{3}=S$$?
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Re: A cylinder has a base with a circumference of 20pi meters  [#permalink]

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Could anyone pl clarify my doubt??

For an equilateral triangle circumscribed in a circle, side = R * root 3 where R is the circumradius.

In this problem, R = 10, So side of the triangle should be 10 * root 3..

Pl point where I am going wrong.
Intern  Joined: 08 Feb 2011
Posts: 11
Re: A cylinder has a base with a circumference of 20pi meters  [#permalink]

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Quote:
Hey Bunuel quick question,

If the radius of a circle that inscribed an equaliteral is $$r=S\sqrt{3}/3$$, where r is the radius and S is the side of the equilateral, should int the answer be $$30/\sqrt{3}=S$$?

I did the same thing as Alphabeta1234 and Friend29. Can someone explain why this is incorrect?
$$S=\frac{30}{\sqrt{3}}$$
$$S=10\sqrt{3}$$.

Thanks
Intern  Joined: 18 Jul 2013
Posts: 31
Re: A cylinder has a base with a circumference of 20pi meters  [#permalink]

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1
C= 20pi =2pi*10
area of circle= pir^2 = pi10^2=100pi
Now as question mentions that probability of dropping a marker anywhere on the base is same and in particular in the equilateral triangle is sqrt3/4. So, equilateral triangle's area is sqrt3/4 of total area of circle (100pi) = sqrt3/4 *100pi( for e.g we do 20% of 100).
Now area of equilateral triangle can also be represented = as per its formula = a^2*sqrt3/4
So now when will be equating both the equations
a^2*sqrt3/4 =sqrt3/4 *100pi
a^2 = 100 pi
a= sqrt 10pi
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Re: A cylinder has a base with a circumference of 20pi meters  [#permalink]

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Bunuel wrote:
Joy111 wrote:
A cylinder has a base with a circumference of 20pi meters and an equilateral triangle inscribed on the interior side of the base. A marker is dropped into the tank with an equal probability of landing on any point on the base. If the probability of the marker landing inside the triangle is (Sqrt 3)/4 , what is the length of a side of the triangle?

A. 3(sqrt 2pi)
B. 3(sqrt 3pi)
C. 10 pi
D. 10(sqrt 3pi)
E. 20 pi

Since the probability of the marker landing on the portion of the base inside the triangle is $$\frac{\sqrt{3}}{4}$$ then the portion of the base (circle) inside the triangle must be $$\frac{\sqrt{3}}{4}$$ of the area of the base.

Next: $$circumference=20\pi=2\pi{r}$$ --> $$r=10$$ --> $$area_{base}=\pi{r^2}=100\pi$$;

The area of the equilateral triangle is $$\frac{\sqrt{3}}{4}$$ of the base: $$area_{equilateral}=\frac{\sqrt{3}}{4}*100\pi$$--> also the ares of the equilateral triangle is $$area_{equilateral}=a^2*\frac{\sqrt{3}}{4}$$, where $$a$$ is the length of a side --> $$area_{equilateral}=a^2*\frac{\sqrt{3}}{4}=\frac{\sqrt{3}}{4}*100\pi$$ --> $$a=10{\sqrt\pi}$$.

Similar question to practice: a-cylindrical-tank-has-a-base-with-a-circumference-of-105453.html

Hope it helps.

I cannot seem to understand the highlighted part.

How do we actually relate the given probability to the area of the base?
Math Expert V
Joined: 02 Sep 2009
Posts: 59587
Re: A cylinder has a base with a circumference of 20pi meters  [#permalink]

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earnit wrote:
Bunuel wrote:
Joy111 wrote:
A cylinder has a base with a circumference of 20pi meters and an equilateral triangle inscribed on the interior side of the base. A marker is dropped into the tank with an equal probability of landing on any point on the base. If the probability of the marker landing inside the triangle is (Sqrt 3)/4 , what is the length of a side of the triangle?

A. 3(sqrt 2pi)
B. 3(sqrt 3pi)
C. 10 pi
D. 10(sqrt 3pi)
E. 20 pi

Since the probability of the marker landing on the portion of the base inside the triangle is $$\frac{\sqrt{3}}{4}$$ then the portion of the base (circle) inside the triangle must be $$\frac{\sqrt{3}}{4}$$ of the area of the base.

Next: $$circumference=20\pi=2\pi{r}$$ --> $$r=10$$ --> $$area_{base}=\pi{r^2}=100\pi$$;

The area of the equilateral triangle is $$\frac{\sqrt{3}}{4}$$ of the base: $$area_{equilateral}=\frac{\sqrt{3}}{4}*100\pi$$--> also the ares of the equilateral triangle is $$area_{equilateral}=a^2*\frac{\sqrt{3}}{4}$$, where $$a$$ is the length of a side --> $$area_{equilateral}=a^2*\frac{\sqrt{3}}{4}=\frac{\sqrt{3}}{4}*100\pi$$ --> $$a=10{\sqrt\pi}$$.

Similar question to practice: a-cylindrical-tank-has-a-base-with-a-circumference-of-105453.html

Hope it helps.

I cannot seem to understand the highlighted part.

How do we actually relate the given probability to the area of the base?

$$P = \frac{(favorable)}{(total)} =\frac{(area \ of \ triangle)}{(area \ of \ circle)} = \frac{\sqrt{3}}{4}$$ --> $$(area \ of \ triangle)=(area \ of \ circle)*\frac{\sqrt{3}}{4}$$.
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Re: A cylinder has a base with a circumference of 20pi meters  [#permalink]

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Why do we need the extra information about probability if finding the radius is enough?
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Re: A cylinder has a base with a circumference of 20pi meters  [#permalink]

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holidayhero wrote:
Quote:
Hey Bunuel quick question,

If the radius of a circle that inscribed an equaliteral is $$r=S\sqrt{3}/3$$, where r is the radius and S is the side of the equilateral, should int the answer be $$30/\sqrt{3}=S$$?

I did the same thing as Alphabeta1234 and Friend29. Can someone explain why this is incorrect?
$$S=\frac{30}{\sqrt{3}}$$
$$S=10\sqrt{3}$$.

Thanks

I am looking at the answer the same way. The side of the triangle can directly be found through the radius of the circle.
Why are we doing it the other way? the answer should be 10 root 3. Re: A cylinder has a base with a circumference of 20pi meters   [#permalink] 25 Nov 2019, 06:54
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