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655-705 Level|   Geometry|      
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Bunuel
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The figure shows the design of a mosaic tile in which the four sides 03 Dec 2019, 02:15
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68% (02:41) correct 32% (03:14) wrong based on 98 sessions

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The figure shows the design of a mosaic tile in which the four sides of the square are the diameters of four intersecting semicircles. Small blue stones are to be placed in the shaded regions and will cover 95 percent of the area of these regions. If each side of the square has length 2 feet, approximately how many square feet of the tile will be covered by the blue stones?


A. 0.9
B. 1.5
C. 2.2
D. 2.9
E. 3.2


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The figure shows the design of a mosaic tile in which the four sides 03 Dec 2019, 04:11
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Bunuel

The figure shows the design of a mosaic tile in which the four sides of the square are the diameters of four intersecting semicircles. Small blue stones are to be placed in the shaded regions and will cover 95 percent of the area of these regions. If each side of the square has length 2 feet, approximately how many square feet of the tile will be covered by the blue stones?


A. 0.9
B. 1.5
C. 2.2
D. 2.9
E. 3.2


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Explanation:
Finding the shaded area is equal to the area of the square less the area not shaded.
There are 4 "not shaded" regions.

Since we are given that Side of Square = 2:
Area Square = 2∗2= 4.

We can find the area of 2 "not shaded" regions by calculating the area of the square less two semi-circles (one circle):
Area Circle = πr^2=pi(1)^2 = π

Therefore, the area of 2 "not-shaded" regions is: 4−π

and the area of 4 "not-shaded" regions is:
2∗(4-π)=8-2π

Since we know the area of the "petals" is the area of the square less 4 "not shaded" regions:
4-(8-2π) = 2π-4 = 2.2 IMO-C

Reference for these kind of question with this link: https://www.youtube.com/watch?v=6_YjIAQ2ezA
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The figure shows the design of a mosaic tile in which the four sides 04 Dec 2019, 08:17
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Bunuel

The figure shows the design of a mosaic tile in which the four sides of the square are the diameters of four intersecting semicircles. Small blue stones are to be placed in the shaded regions and will cover 95 percent of the area of these regions. If each side of the square has length 2 feet, approximately how many square feet of the tile will be covered by the blue stones?

A. 0.9
B. 1.5
C. 2.2
D. 2.9
E. 3.2
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Each shaded region is made of 2 equal slices.
\(Diameter=2…Radius=1\)
\(Slice = Area.Sector - Area.Isosceles\)
\(Slice = πr^2*(90/360) - r^2/2=(πr^2-2r^2)/4=(π-2)/4\)
\(Total.Slices=8…Area.Slices=8(π-2)/4=2(1.124)=2.248\)
\(Stones=0.95Total.Slices=aprox(2.2)\)

Ans (C)
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The figure shows the design of a mosaic tile in which the four sides Updated on: 31 Aug 2021, 00:17
Originally posted by Fdambro294 on 31 Aug 2021, 00:12.
Last edited by Fdambro294 on 31 Aug 2021, 00:17, edited 2 times in total.
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(1st) we can drawn in only 2 of the identical semicircles with diameter = 2. Have these be vertically opposite from each other.

The two semicircles will intersect in the center of the square and the point of Tangency will be the exact center of the Square (b/c vertically opposite each other)

The area OUTSIDE these 2 vertically opposite semicircles will be TWO of the “triangular like” white areas outside the bounds of the 2 semicircles but inside the square.

this means: (Area of Square) - (Area of 2 semicircles) = (2 of the white triangular-like” Areas inside the Square)

2 identical semi circles with diameter = 2 (r = 1) is the same as ONE Whole Circle with radius = 1

(2)^2 - (1)^2 (pi) = area of 2 of the white triangular-like sections inside the Square =

(4 - (pi))



(2nd)To get the Area of ALL FOUR of these white triangular sections, we need to double the area for TWO we found above:

(2) * (4 - (pi)) = Area of White “triangular-like” sections inside the square

(3rd) to find the area of the shaded portions covered, we need to subtract the above Area from the entire Area of the Square and then take 95% of that value

95% * [ (2)^2 - (2) * (4 - (pi)) ]

We can estimate (pi) at around ~ 3.1

95% * [ (4) - (2) * (4 - 3.1) ]

95% * [ (4) - (2) * (.9) ]

95% * [ 2.2 ]

Since we have underestimated the value of (pi) ——-> the value of 2.2 will be slightly overvalued.


(C) 2.2 is the approximate value

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The figure shows the design of a mosaic tile in which the four sides 31 Aug 2021, 00:14
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Solution:

The approach is simple.

Area of shaded region = Area if 4 semi circles - Area of square.

When we calculate the area of the 4 semi-circles, we calculate the shaded region twice and the unshaded region once.

Now if you look close, this extra area (1 part of shaded + 1 part of unshaded) is nothing but area of the square.

So, Area of shaded region \(= 4\times \frac{\pi \times 1^2}{2} - (2\times 2)\)

We know radius os semicircle \(= 1\) and its area \(= \frac{\pi \times radius^2}{2}\)

\(⇒ 2\pi-4\)

\(⇒ 2\times \frac{22}{7} - 4\)

\(⇒ 2.2 \)

Hence the right answer is Option C.
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The figure shows the design of a mosaic tile in which the four sides 31 Aug 2021, 00:24
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Find the area of one small piece and multiply by 8 to get the shaded area and then 95% to get the area

8*(Area of sector - Area of triangle)*95% = 8*(Pi/4-1/2)*95% = (2Pi-4)*95% = approx 2.2
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