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# The figure shows the design of a mosaic tile in which the four sides

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Math Expert
Joined: 02 Sep 2009
Posts: 60687
The figure shows the design of a mosaic tile in which the four sides  [#permalink]

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03 Dec 2019, 02:15
00:00

Difficulty:

55% (hard)

Question Stats:

71% (02:47) correct 29% (03:31) wrong based on 34 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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1.jpg [ 19.11 KiB | Viewed 697 times ]

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Re: The figure shows the design of a mosaic tile in which the four sides  [#permalink]

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03 Dec 2019, 04:11
1
Bunuel wrote:

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

Are You Up For the Challenge: 700 Level Questions

Attachment:
1.jpg

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

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Re: The figure shows the design of a mosaic tile in which the four sides  [#permalink]

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04 Dec 2019, 08:17
Bunuel wrote:

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

$$Diameter=2…Radius=1$$
$$Slice = Area.Sector - Area.Isosceles$$
$$Slice = πr^2*(90/360) - r^2/2=(πr^2-2r^2)/4=(π-2)/4$$
$$Total.Slices=8…Area.Slices=8(π-2)/4=2(1.124)=2.248$$
$$Stones=0.95Total.Slices=aprox(2.2)$$

Ans (C)
Re: The figure shows the design of a mosaic tile in which the four sides   [#permalink] 04 Dec 2019, 08:17
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