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The surface of a mirror is composed of a rectangular piece that is 9 [#permalink]
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31 Jul 2016, 01:08
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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. Attachment:
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Re: The surface of a mirror is composed of a rectangular piece that is 9 [#permalink]
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31 Jul 2016, 01:34
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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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Re: The surface of a mirror is composed of a rectangular piece that is 9 [#permalink]
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31 Jul 2016, 01:40
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.



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Re: The surface of a mirror is composed of a rectangular piece that is 9 [#permalink]
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31 Jul 2016, 01:44
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. Every step taken with caution will lead you closer to your dream score.
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Re: The surface of a mirror is composed of a rectangular piece that is 9 [#permalink]
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03 Aug 2016, 10:08
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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Re: The surface of a mirror is composed of a rectangular piece that is 9 [#permalink]
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03 Aug 2016, 10:43
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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Re: The surface of a mirror is composed of a rectangular piece that is 9 [#permalink]
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26 Aug 2016, 16:05
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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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Re: The surface of a mirror is composed of a rectangular piece that is 9 [#permalink]
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26 Aug 2016, 20:32
Another approach:
First, observe that the radius of the semicircle 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!



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Re: The surface of a mirror is composed of a rectangular piece that is 9 [#permalink]
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26 Aug 2016, 20:57
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stonecold wrote: Donnie84 wrote: Another approach:
First, observe that the radius of the semicircle 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.



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Re: The surface of a mirror is composed of a rectangular piece that is 9 [#permalink]
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26 Aug 2016, 21:16
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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.



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