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The figure shown above consists of three identical circles that are tangent to each other. If the area of the shaded region is \(64\sqrt{3}-32\pi\), what is the radius of each circle?
(A) 4
(B) 8
(C) 16
(D) 24
(E) 32
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The figure shown above consists of three identical circles that are tangent to each other. If the area of the shaded region is \(64\sqrt{3}-32\pi\), what is the radius of each circle?
(A) 4
(B) 8
(C) 16
(D) 24
(E) 32
Let the radius of the circle be \(r\), then the side of equilateral triangle will be \(2r\).
Area of the shaded region equals to area of the equilateral triangle minus area of three 60 degrees sectors.
Area of a 60 degree sector is 1/6 of the are of whole circle (as whole circle is 360 degrees and 60 is 1/6 of it), hence the area of 3 such sectors will be 3/6=1/2 of the area of whole circle, so \(area_{sectors}=\frac{\pi{r^2}}{2}\) (here if you could spot that \(\frac{\pi{r^2}}{2}\) should correspond to \(32\pi\) then you can write \(\frac{\pi{r^2}}{2}=32\pi\) --> \(r=8\));
Area of equilateral triangle equals to \(a^2\frac{\sqrt{3}}{4}\), where \(a\) is the length of a side. So in our case \(area_{equilateral}=(2r)^2*{\frac{\sqrt{3}}{4}}=r^2\sqrt{3}\);
Area of the shaded region equals to \(64\sqrt{3}-32\pi\), so:
- \(area_{equilateral}-area_{sectors}=r^2\sqrt{3}-\frac{\pi{r^2}}{2}=64\sqrt{3}-32\pi\);
\(r^2=\frac{2(64\sqrt{3}-32\pi)}{2\sqrt{3}-\pi}=\frac{64(2\sqrt{3}-\pi)}{(2\sqrt{3}-\pi)}=64\);
\(r=8\).
Answer: B.
GEOMETRY: Shaded Region Problems: https://gmatclub.com/forum/geometry-sha ... 75005.html
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Find the are of equilateral triangle with side as 2r
Find the are of the three sectors (eq. trianlge has \(\angle\) = \(60^{\circ}\) )
Subtract both to get the area of shaded region. You will get the equation as
\(\sqrt{3}r^{2} - \frac{ \pi r^{2}}{2} = 64\sqrt{3} - 32 \pi\)
Don't even need to solve the equation you can see \(r^{2} = 64\) OR \(\frac{r^{2}}{2} = 32\)
Hence, r = 8
Answer - B
Time Taken - 2:00
Difficulty level - 700
Find the are of the three sectors (eq. trianlge has \(\angle\) = \(60^{\circ}\) )
Subtract both to get the area of shaded region. You will get the equation as
\(\sqrt{3}r^{2} - \frac{ \pi r^{2}}{2} = 64\sqrt{3} - 32 \pi\)
Don't even need to solve the equation you can see \(r^{2} = 64\) OR \(\frac{r^{2}}{2} = 32\)
Hence, r = 8
Answer - B
Time Taken - 2:00
Difficulty level - 700
General Discussion
PareshGmat
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Updated on: 16 Oct 2014, 18:17
Originally posted by PareshGmat on 09 Mar 2014, 17:55.
Last edited by PareshGmat on 16 Oct 2014, 18:17, edited 2 times in total.
Last edited by PareshGmat on 16 Oct 2014, 18:17, edited 2 times in total.
Let the radius of each circle = r
The triangle formed is a equilateral triangle with each side = 2r
So Area of the Triangle = \(\frac{\sqrt{3}}{4} * (2r)^2 = \sqrt{3} r^2\) ....................... (1)
This triangle consists of 3 sectors each of 60 Degrees i.e total 180 Deg
Area of these sectors = \(\frac{\pi r^2}{2}\) .................. (2)
Area of shaded region = (1) - (2)
\(\sqrt{3}r^2 - \frac{\pi r^2}{2} = 64\sqrt{3} - 32\pi\)
Solving the above, radius r = 8 = Answer = B
The triangle formed is a equilateral triangle with each side = 2r
So Area of the Triangle = \(\frac{\sqrt{3}}{4} * (2r)^2 = \sqrt{3} r^2\) ....................... (1)
This triangle consists of 3 sectors each of 60 Degrees i.e total 180 Deg
Area of these sectors = \(\frac{\pi r^2}{2}\) .................. (2)
Area of shaded region = (1) - (2)
\(\sqrt{3}r^2 - \frac{\pi r^2}{2} = 64\sqrt{3} - 32\pi\)
Solving the above, radius r = 8 = Answer = B
Option B.
The triangle formed by the circles will be an equilateral triangle.
Area of triangle=\(\sqrt{3}/4*(side)^2\)
And side =2*radius
Now area of triangle=area shaded+area of 3 sectors of the circles.
Area shaded=given
Area of 3 sectors=\(3*[60/360*pi*r^2]\)
We'll equate the 2 sides[Area triangle=Area shaded+area of 3 sectors]
and simplify to get r=8
The triangle formed by the circles will be an equilateral triangle.
Area of triangle=\(\sqrt{3}/4*(side)^2\)
And side =2*radius
Now area of triangle=area shaded+area of 3 sectors of the circles.
Area shaded=given
Area of 3 sectors=\(3*[60/360*pi*r^2]\)
We'll equate the 2 sides[Area triangle=Area shaded+area of 3 sectors]
and simplify to get r=8
Is this a valid way of approaching the question, or just a lucky coincidence:
Area of equilateral triangle is \((S^2\sqrt{3})/4\)
So \((S^2\sqrt{3})/4\) = \(64\sqrt{3}\)
Simplify this to\(S = 16\sqrt{3}/\sqrt{3}\) -> = 16. One side is 16, divide by 2 to get 8.
I simply ignored the "-32" - is this a valid way of doing it?
Area of equilateral triangle is \((S^2\sqrt{3})/4\)
So \((S^2\sqrt{3})/4\) = \(64\sqrt{3}\)
Simplify this to\(S = 16\sqrt{3}/\sqrt{3}\) -> = 16. One side is 16, divide by 2 to get 8.
I simply ignored the "-32" - is this a valid way of doing it?
Posts: 3,651
erikvm
Hi erikvm,
The best approach would be to write down the equations for the area of shaded region and then equate both sides. Your method worked because the final expression was written in the form where you could equate two expressions from LHS and RHS. This may not be the case always and hence is not the best practice to follow.
Also to answer bgpower's question, the final equation can be written as
\(\sqrt{3}r^2 - \frac{πr^2}{2} = 64\sqrt{3} - 32π\)
There are two easy ways to solve the equation for r.
Equating the expression
You can see the similarity in expression on both sides of the equation and can equate \(\frac{πr^2}{2} = 32π\) or \(\sqrt{3}r^2 = 64\sqrt{3}\) . This would give you the value of \(r = 8\).
Solving the equation
Alternatively the equation can also be solved very quickly. You just need to be aware of the possibilities of the terms canceling out.
The equation can be written as \(r^2 =\) (\(64\sqrt{3} - 32π)/(\sqrt{3}- \frac{π}{2})\)
If we take \(64\) common from the numerator we will have the expression as \(r^2 =\) \(64 *( \sqrt{3} - \frac{π}{2})/(\sqrt{3}- \frac{π}{2})\) thus resulting in \(r^2 = 64\) and \(r = 8\)
So with the above approach, it is possible to solve the question in less than 2 minutes.
Hope this helps
Regards
Harsh
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I used guesstimation which works for people who suck at math like me.
All you need to know is √3 = 1.7
pi=3.14
64*1.7 - 32*3.14
Is more or less 8
The shaded region is about (1/4) of the triangle so
a^2(√3/4)=4*8
a^2 = 32*(4/√3)
a^2 = 128/1.7 is less than 260 but more than 192
16*16=256
so r = 16/2 = 8
Took me 5 minutes though
All you need to know is √3 = 1.7
pi=3.14
64*1.7 - 32*3.14
Is more or less 8
The shaded region is about (1/4) of the triangle so
a^2(√3/4)=4*8
a^2 = 32*(4/√3)
a^2 = 128/1.7 is less than 260 but more than 192
16*16=256
so r = 16/2 = 8
Took me 5 minutes though
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Expert reply
Attached is a visual that should help.
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Bunuel
We can let each radius = r, and so the side of each triangle = 2r.
Notice that the area of the equilateral triangle consists of the central shaded region and three identical circular sectors, each of which is a 60-degree sector from its circle. Using this information, we can create the following equation:
(Area of equilateral triangle) - (3 x area of 1/6 of each circle) = area of shaded region
(2r)^2√3/4 - 3(1/6 x π r^2) = 64√3 − 32π
[(4r^2)√3]/4 - (πr^2)/2 = 64√3 − 32π
(r^2)√3 - (πr^2)/2 = 64√3 − 32π
Multiplying both sides by 2, we have:
2(r^2)√3 - πr^2 = 128√3 − 64π
r^2(2√3 - π) = 128√3 − 64π
Dividing both sides by (2√3 - π), we have:
r^2 = 64
r = 8
Answer: B
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There is a very simple of way of solving this.
Basically the equation is nothing but the Area of the Triangle - The 3 Sectors = 64\sqrt{3}-32π
So area of the equilateral triangle is 64\sqrt{3}
Area of an equilateral triangle = side^2*\sqrt{3}/4
side^2=64*4
side=16
Radius is 16/2=8.
Answer is B.
Basically the equation is nothing but the Area of the Triangle - The 3 Sectors = 64\sqrt{3}-32π
So area of the equilateral triangle is 64\sqrt{3}
Area of an equilateral triangle = side^2*\sqrt{3}/4
side^2=64*4
side=16
Radius is 16/2=8.
Answer is B.
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Bunuel
Hello,
can you help me understand how did we arrive at pi*r^2/2? Usually the area of the sector formula is angle/360* pi*r^2
Jaya6
We have three 60-degree sectors. Together they make 180-dgeree sector, or half the circle, which is 360-degree. The area of half a circle is pi*r^2/2.
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Area of equilateral triangle is given by \(\frac{s^2\sqrt{3}}{4}\), where s is the length of a side.
In this case, the length of the side is 2r. S = 2r -->
\(\frac{(2r)^2\sqrt{3}}{4}\) = \(r^2\sqrt{3}\)
Now we know that the irrational terms in the expression for the area is not likely to be affected by each other: \(64\sqrt{3} - 32Pi\)
So we can conclude that r^2 = 64.
In this case, the length of the side is 2r. S = 2r -->
\(\frac{(2r)^2\sqrt{3}}{4}\) = \(r^2\sqrt{3}\)
Now we know that the irrational terms in the expression for the area is not likely to be affected by each other: \(64\sqrt{3} - 32Pi\)
So we can conclude that r^2 = 64.
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I have taken shortcut to get the answer, which takes 1-2 min to solve.
since they are just asking for the radius of circle.
If you see 64√3 represents area of triangle , whereas 32 pi represents Area of 3 (1/6)th of a circle .
taking just 32 pi ( 3*1/6 =1/2) gives us half Area of circle , multiple by 2 gives 64 pi .
therefore , Area of circle 64 pi and radius is 8.
since they are just asking for the radius of circle.
If you see 64√3 represents area of triangle , whereas 32 pi represents Area of 3 (1/6)th of a circle .
taking just 32 pi ( 3*1/6 =1/2) gives us half Area of circle , multiple by 2 gives 64 pi .
therefore , Area of circle 64 pi and radius is 8.
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Updated on: 10 Jun 2022, 14:05
Originally posted by ThatDudeKnows on 13 May 2022, 11:42.
Last edited by ThatDudeKnows on 10 Jun 2022, 14:05, edited 1 time in total.
Last edited by ThatDudeKnows on 10 Jun 2022, 14:05, edited 1 time in total.
Bunuel
Why are y'all doing so much math?
The shaded area is the area of the dashed triangle minus three little pie slices. The part we're subtracting is 32(pi), so that's the pie slices. Just eyeballing it, we can see that the three pie slices add up to half of one of the circles (yeah, we know this for sure because each of the angles of the dashed triangle is 60 degrees, so each slice is 1/6th of a circle since 60 is 1/6th of 360...but we don't need to even know this to figure it out, since we can just eyeball it).
Anyway, half of the area of a circle = 32(pi), so the area of a circle is 64(pi). A=(pi)r^2. So r=8.
Answer choice B.
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MartyTargetTestPrep
Silly question... why do you have to multiply both sides of the equation by 2? To confirm, whatever you to do one side of the equation, you have to do to the other side, correct? Happy Holidays
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woohoo921
Yes, you have to do the same thing to both sides.
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once we get to (2r)^2√3/4 - 3(1/6 x π r^2) = 64√3 − 32π
[(4r^2)√3]/4 - (πr^2)/2 = 64√3 − 32π
(r^2)√3 - (πr^2)/2 = 64√3 − 32π
can we just say r^2)√3 = 64√3
and therefore r = 8? or do we need to multiply both sides by 2 and follow the remaining steps
Thank you
once we get to (2r)^2√3/4 - 3(1/6 x π r^2) = 64√3 − 32π
[(4r^2)√3]/4 - (πr^2)/2 = 64√3 − 32π
(r^2)√3 - (πr^2)/2 = 64√3 − 32π
can we just say r^2)√3 = 64√3
and therefore r = 8? or do we need to multiply both sides by 2 and follow the remaining steps
Thank you
MartyTargetTestPrep
Multiplying both sides by 2, we have:
2(r^2)√3 - πr^2 = 128√3 − (2√3 - π)
r^2(2√3 - π) = 128√3 − 64π. ((((1))))
Dividing both sides by (2√3 - π), we have:
r^2 = 64. ((((2)))
r = 8
Answer: B
how do we get from ((((1)))) to ((((2)))?
wouldnt that become (128√3 − 64π) / (2√3 - π), = 64 - 64? little confused with the math behind this step sorry
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