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Current Student B
Joined: 08 Feb 2016
Posts: 69
Location: India
Concentration: Technology
Schools: AGSM '20 (A)
GMAT 1: 650 Q49 V30 GPA: 4
The surface of a mirror is composed of a rectangular piece that is 9  [#permalink]

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1
10 00:00

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(N/A)

Question Stats: 67% (01:52) correct 33% (02:07) wrong based on 203 sessions

### HideShow timer Statistics The surface of a mirror is composed of a rectangular piece that is 9 feet long and two semicircular pieces whose diameters are equal to the width of the rectangular piece, as shown in figure. If the ratio of the area of the rectangular piece to the total area of the two semicircular pieces is 9/pi , what is the width of the rectangular piece, in feet?

A. 1
B. 2
C. 3
D. 4
E. 5

I solved it this way.:
Area of rectangular piece = 9b ( Let b = width of rectangle = diameter of the 2 semicircles)
Area of 1 semicircular piece = pi.(D^2)/4
Area of 2 semicircular pieces = pi.(D^2)/2

So, putting the ratio = 9/pi, I get D = b =2 -> which is not the OA.

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Current Student D
Joined: 12 Aug 2015
Posts: 2617
Schools: Boston U '20 (M)
GRE 1: Q169 V154 Re: The surface of a mirror is composed of a rectangular piece that is 9  [#permalink]

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4
1
Here is my approach
Let the width be x
here the two semicircles make the entire uniform circle of radius pie*x^2/4
area of rect=> 9*x
Given => 9x/pie*x^2/4 = 9/pie => x=4
SMASH THAT D
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Board of Directors V
Status: Stepping into my 10 years long dream
Joined: 18 Jul 2015
Posts: 3631
Re: The surface of a mirror is composed of a rectangular piece that is 9  [#permalink]

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2
ajay2121988 wrote:
The surface of a mirror is composed of a rectangular piece that is 9 feet long and two semicircular pieces whose diameters are equal to the width of the rectangular piece, as shown in figure. If the ratio of the area of the rectangular piece to the total area of the two semicircular pieces is 9/pi , what is the width of the rectangular piece, in feet?

A. 1
B. 2
C. 3
D. 4
E. 5

I solved it this way.:
Area of rectangular piece = 9b ( Let b = width of rectangle = diameter of the 2 semicircles)
Area of 1 semicircular piece = pi.(D^2)/4
Area of 2 semicircular pieces = pi.(D^2)/2

So, putting the ratio = 9/pi, I get D = b =2 -> which is not the OA.

Area of 1 semicircular piece will be equal to 1/2 * pi.(D^2)/4 and not pi.(D^2)/4.

When you will equate 9b/(1/2 * pi.(D^2)/4) = 9/pi, it will give b = 4.
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Current Student B
Joined: 08 Feb 2016
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Concentration: Technology
Schools: AGSM '20 (A)
GMAT 1: 650 Q49 V30 GPA: 4
Re: The surface of a mirror is composed of a rectangular piece that is 9  [#permalink]

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abhimahna wrote:
ajay2121988 wrote:
The surface of a mirror is composed of a rectangular piece that is 9 feet long and two semicircular pieces whose diameters are equal to the width of the rectangular piece, as shown in figure. If the ratio of the area of the rectangular piece to the total area of the two semicircular pieces is 9/pi , what is the width of the rectangular piece, in feet?

A. 1
B. 2
C. 3
D. 4
E. 5

I solved it this way.:
Area of rectangular piece = 9b ( Let b = width of rectangle = diameter of the 2 semicircles)
Area of 1 semicircular piece = pi.(D^2)/4
Area of 2 semicircular pieces = pi.(D^2)/2

So, putting the ratio = 9/pi, I get D = b =2 -> which is not the OA.

Area of 1 semicircular piece will be equal to 1/2 * pi.(D^2)/4 and not pi.(D^2)/4.

When you will equate 9b/(1/2 * pi.(D^2)/4) = 9/pi, it will give b = 4.

Oh, when will I stop doing these silly mistakes !

Anyways, thanks, K +1.
Board of Directors V
Status: Stepping into my 10 years long dream
Joined: 18 Jul 2015
Posts: 3631
Re: The surface of a mirror is composed of a rectangular piece that is 9  [#permalink]

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1
ajay2121988 wrote:
abhimahna wrote:
ajay2121988 wrote:
The surface of a mirror is composed of a rectangular piece that is 9 feet long and two semicircular pieces whose diameters are equal to the width of the rectangular piece, as shown in figure. If the ratio of the area of the rectangular piece to the total area of the two semicircular pieces is 9/pi , what is the width of the rectangular piece, in feet?

A. 1
B. 2
C. 3
D. 4
E. 5

I solved it this way.:
Area of rectangular piece = 9b ( Let b = width of rectangle = diameter of the 2 semicircles)
Area of 1 semicircular piece = pi.(D^2)/4
Area of 2 semicircular pieces = pi.(D^2)/2

So, putting the ratio = 9/pi, I get D = b =2 -> which is not the OA.

Area of 1 semicircular piece will be equal to 1/2 * pi.(D^2)/4 and not pi.(D^2)/4.

When you will equate 9b/(1/2 * pi.(D^2)/4) = 9/pi, it will give b = 4.

Oh, when will I stop doing these silly mistakes !

Anyways, thanks, K +1.

Thanks.

Simple strategy to avoid silly mistakes is to get completely immersed in the question you are doing.

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Senior Manager  Joined: 11 Nov 2014
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Re: The surface of a mirror is composed of a rectangular piece that is 9  [#permalink]

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L*B = area of rectangle
pir^2/2 = area of semi circle

L*B + 2(pir^2/2) = 9/pi
9*x + 2(pix^2/2) = 9/pi

how do I proceed please?
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Status: Stepping into my 10 years long dream
Joined: 18 Jul 2015
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Re: The surface of a mirror is composed of a rectangular piece that is 9  [#permalink]

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paidlukkha wrote:
L*B = area of rectangle
pir^2/2 = area of semi circle

L*B + 2(pir^2/2) = 9/pi
9*x + 2(pix^2/2) = 9/pi

how do I proceed please?

Its is the ratio between the rectangle and semi circles given and NOT the sum.

Your equation will be

9*x / 2(pix^2/2) = 9/pi

Now proceed and let me know if you need more help!!
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Manager  Joined: 04 Jan 2014
Posts: 117
GMAT 1: 660 Q48 V32 GMAT 2: 630 Q48 V28 GMAT 3: 680 Q48 V35 Re: The surface of a mirror is composed of a rectangular piece that is 9  [#permalink]

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Another approach:

First, observe that the radius of the semi-circle and the combined circle will be half of width of the rectangle. Next, as per the data given in the question and answer choices, the width should be a multiple of 2. Options A, C and E are ruled out.

Let's check with option B.
If width = 2, area of rectangle = 9*2 = 18
Radius of the circle = 1
Area of the circle = pi

Ratio = 18/pi -> ruled out since we want the ration to be 9/pi.

Answer (D).

Validation:

If width = 4, area of rectangle = 9*4 = 36
Radius of the circle = 2
Area of the circle = 4*pi

Ratio = 9/pi -> good!
Manager  Joined: 04 Jan 2014
Posts: 117
GMAT 1: 660 Q48 V32 GMAT 2: 630 Q48 V28 GMAT 3: 680 Q48 V35 Re: The surface of a mirror is composed of a rectangular piece that is 9  [#permalink]

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1
stonecold wrote:
Donnie84 wrote:
Another approach:

First, observe that the radius of the semi-circle and the combined circle will be half of width of the rectangle. Next, as per the data given in the question and answer choices, the width should be a multiple of 2. Options A, C and E are ruled out.=> This is same as doing the long calculation. And width being a multiple of 2 is known only after the initial step.
Let's check with option B.
If width = 2, area of rectangle = 9*2 = 18
Radius of the circle = 1
Area of the circle = pi

Ratio = 18/pi -> ruled out since we want the ration to be 9/pi.

Answer (D).

Validation:

If width = 4, area of rectangle = 9*4 = 36
Radius of the circle = 2
Area of the circle = 4*pi

Ratio = 9/pi -> good!

I meant to say that the width must be even. Width cannot be odd in this case because that would mean a fraction value for the radius and its square would be messy. The values in the question are elegant, so we can eliminate the options with width = 1 or 3 or 5.
Manager  Joined: 04 Jan 2014
Posts: 117
GMAT 1: 660 Q48 V32 GMAT 2: 630 Q48 V28 GMAT 3: 680 Q48 V35 Re: The surface of a mirror is composed of a rectangular piece that is 9  [#permalink]

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1
Yes, that is my opinion and I'm sticking to it. Intuition told me that I should start with options B and D. We are here to learn from each other and improve. I take your feedback constructively.
Intern  B
Joined: 15 Sep 2018
Posts: 9
Re: The surface of a mirror is composed of a rectangular piece that is 9  [#permalink]

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stonecold wrote:
Here is my approach
Let the width be x
here the two semicircles make the entire uniform circle of radius pie*x^2/4
area of rect=> 9*x
Given => 9x/pie*x^2/4 = 9/pie => x=4
SMASH THAT D

Could someone please explain why we have to divide by 4? "9x/pie*x^2/4"

Thanks!
Intern  B
Joined: 16 Nov 2015
Posts: 22
Location: United Kingdom
Re: The surface of a mirror is composed of a rectangular piece that is 9  [#permalink]

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Aviv29 Don't worry I got confused to and it was a silly mistake on my end. Once I explain, you will have an Aha moment

We've been given the length of rectangle = 9
Width of rectangle, lets make it = x

As it mentions that the width of the rectangle is the same as the diameter of the semicircle, to get the radius, you have to divide x by 2.

Therefore, radius of the semicircle = x/2

As area of a circle = pi*r^2, so now lets square the radius of the semicircle and multiply by pi

Area of the two semicircle = pi * ((x/2)^2) = pi * (x^2)/4

Now you can do the ratio.

Given => 9x /pi*(x^2/4) = 9/pi => x=4

Hope this makes sense. Re: The surface of a mirror is composed of a rectangular piece that is 9   [#permalink] 20 Apr 2019, 13:18
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# The surface of a mirror is composed of a rectangular piece that is 9

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