Kaplan 800 - Geometry

Question

#9 - Pg 319. Kaplan 800.

If each curved portion of the boundary of the figure above is formed form the circumferences of two semicircles, each with a radius of 2, and each of the parallel sides has a length 4, what is the area of the shaded figure?

They have a diagram the best way to describe it is to imagine bending a sheet of paper so that it forms one arch on the top and one on the bottom. The resulting troughs will form two unclosed semicircles.

a:16
b: 32
c: 16-8pi
d: 32-8pi
e: 32-4pi

Spoiler, my question/logic is below.

The way I approached this problem was I used the radius to find the circumference of the two semicircles. So I took the radius of 2*2pi = 4pi. Because I only have a semicircle, I divided that by 2 to get 2pi. Because there are two troughs/semicircles multiply by 2 to get back to 4pi. To find the total area, I said 4pi*4 = 16pi, which is not an answer. 
Kaplan says you just take the radius length of the two semicircles multiplied by the width, so (2+2+2+2)*4=32.

Since the edge of the bent paper is half of the circumference, I thought you had to use that formula. Additionally, you cannot assume that diameter = 1/2 of the circumference.

Any insight is appreciated.

fivedaysleft · Answer

post the diagram please

jamifahad · Answer

Diagram please!!
Never mind.
A quick google search reveals this diagram for this question

Attachment:

Screen shot 2011-07-03 at 10.02.56 PM.png [ 23.68 KiB | Viewed 2388 times ]

I got confused at first too. Take any A4 paper and try to make the shape same as in diagram. You will realize that area of curved surface is nothing but the total area of that rectangular sheet.

Area= length of rectangular sheet * width of rectangular sheet
= 2*4 * 4
=32.

OA B.

Pique · Answer

I understand the concept of the shape being a rectangle, but I find that the information provided does not coincide with what it implies.

Starting at 32. 32=8*4. The curved edge depicted then, the length of the piece of paper, is 8. 
Imagine if were to tape a piece of paper together so that it formed a cylinder with the long edge forming the circumference. In this case, the side length of 8 would correlate to a circumference of 8. The reason we can use the circumference formula is because we have two equal semi-circles. 
Now we know the formula for circumference is 2*pi*r. So, 8=2*pi*r. This results with 4=pi*r and to the result is that the radius is 4/pi. 4/pi is not what the problem provided for the radius (it said 2).

If we use the radius of 2*4 we are drawing straight through open space, rather than fully tracing and finding the edge of the piece of paper.

Does this make sense, or am I off track somewhere?

fluke · Answer

I am getting  16 \pi (Area of the rectangle w=4, l=2*2*pi*2/2=4*pi OR the lateral surfaced area of a cylinder with radius 2 and height 4). I couldn't understand what the question is trying to convey.

jamifahad · Answer

If you look at the diagram, you would notice that it is the "lateral" surface area of cylinder.
Ex:- Rectangular sheet of 8x10 has area of 80. When that sheet is folded such that 8 becomes circumference and 10 becomes height, then the lateral surface area of cylinder formed will be 2 pi r h. 
Now 8=2 pi r 
r= 4/pi
Hence lateral surface= 2 pi r h = 2 pi 4/pi 10=80. Hence lateral surface area is nothing but the area of rectangular sheet.

fivedaysleft · Answer

no disrespect but i think with cylinders matters are getting comlicated. my approach is graphic, for a layman (like me) :-D

if you see the attachment below, it can be seen that the semicircles can be cut, inverted and pasted in the existing troughs ( holes ). after cutting and fitting we get length = 2r + 2r = 4r and width = r+r = 2r multiply and get area = 8r*r and r = 2 so area = 8*4 = 32

Warlock007 · Answer

Even I am getting 16 \pi with the same approach any clarification further????????

Kaplan 800 - Geometry

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