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What is the area of the region of diagram?

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by Max@Math Revolution » Mon May 02, 2016 10:46 pm
What is the area of the region of diagram?

A. 8+(3Ï€/2)
B. 10+(3Ï€/2)
C. 8+3Ï€
D. 10+3Ï€
E. 8+6Ï€

* A solution will be posted in two days.
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by Max@Math Revolution » Sun May 08, 2016 5:55 pm
As you can see from the figure attached, we have a rectangle that has an area of 4*2=8 and 2*1=2. Also, there are three semicircles with 1 as their radii. All together, the area is 8+2+3(Ï€/2)= 10+(3Ï€/2). Hence, B is the correct answer.
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by 800_or_bust » Sun May 08, 2016 7:26 pm
Max@Math Revolution wrote:What is the area of the region of diagram?

A. 8+(3Ï€/2)
B. 10+(3Ï€/2)
C. 8+3Ï€
D. 10+3Ï€
E. 8+6Ï€

* A solution will be posted in two days.
First let's determine the area of each semi-circular region. The area of a circle is equal to π(r^2). Here, r=1 so the area of any given full circle of radius 1 would be equal to π. However, we have three semi (half) circles. Therefore, the area of each semi-circle is π/2. And since we have 3 of them, the total area of the semicircular regions is 3π/2.

Next, turning to the rectangular region. The larger rectangle at top is 4 units in width, and the length is 2 units (equal to the diameter of the semicircles). So, the area of this rectangle is 8. The smaller rectangle has width of 2 units (equal to the diameter of the semicircle). The length of this rectangle is a little more challenging, as the line segment of length 2 extends into the upper rectangular region. So we need to subtract the radius of the semicircle to avoid double counting. Therefore, the uncounted length is equal to 1. And the area of this rectangle is simply 2.

Adding up the two rectangles and three semicircles, we get 10+3Ï€/2.
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