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# A circular lawn with a radius of 5 meters is surrounded by a circular

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Manager
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A circular lawn with a radius of 5 meters is surrounded by a circular  [#permalink]

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Updated on: 06 Oct 2018, 03:19
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5% (low)

Question Stats:

87% (00:49) correct 13% (00:53) wrong based on 95 sessions

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A circular lawn with a radius of 5 meters is surrounded by a circular walkway that is 4 meters wide. What is the area of the walkway?

A. 81 π
B. 56 π
C. 25 π
D. 20 π
E. cannot be determined

To figure out the area of the walkway I found the difference between of the larger circle (circular lawn + circular walkway). To find the radius of the bigger circle I added the diameter of both the circular walkway and the circular lawn and divided by 2. (10+4)/2. I found the radius of the circle to be 7. So area of the bigger circle would be (7)squared times pi.

The Manhattan guide explains that the area of the bigger circle is 9(squared) times pie. Why is the radius of the bigger circle 9? I understand that the radius of the smaller circle is 5 but wouldn't the radius of the circular walkway be 2 since the entire walkway is 4 meters wide?

Jose

Originally posted by josemnz83 on 22 Jun 2013, 07:35.
Last edited by Bunuel on 06 Oct 2018, 03:19, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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Re: A circular lawn with a radius of 5 meters is surrounded by a circular  [#permalink]

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22 Jun 2013, 09:14
josemnz83 wrote:
A circular lawn with a radius of 5 meters is surrounded by a circular walkway that is 4 meters wide. What is the area of the walkway?

To figure out the area of the walkway I found the difference between of the larger circle (circular lawn + circular walkway). To find the radius of the bigger circle I added the diameter of both the circular walkway and the circular lawn and divided by 2. (10+4)/2. I found the radius of the circle to be 7. So area of the bigger circle would be (7)squared times pi.

The Manhattan guide explains that the area of the bigger circle is 9(squared) times pie. Why is the radius of the bigger circle 9? I understand that the radius of the smaller circle is 5 but wouldn't the radius of the circular walkway be 2 since the entire walkway is 4 meters wide?

Jose

There is a figure attached to the original question. See below:
Attachment:

Circle.png [ 6.23 KiB | Viewed 10986 times ]
A circular lawn with a radius of 5 meters is surrounded by a circular walkway that is 4 meters wide (see figure). What is the area of the walkway?

Hope it heps.
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Re: A circular lawn with a radius of 5 meters is surrounded by a circular  [#permalink]

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22 Jun 2013, 09:36
Thanks Bunuel. I have each of the Manhattan guides so I had seen the figure before. I did not post it because I did not want to violate posting rules. Anyway even after looking at the figure I cannot understand why the radius of the walkway is not 2. Radius is half of the diameter. The problem says that the walkway is 4 meters wide. Is 4 meters not the diameter? Or is the radius 4 because the walkway is not really a "full circle" but more like a semicircle of sorts?

Thanks!

Bunuel wrote:
josemnz83 wrote:
A circular lawn with a radius of 5 meters is surrounded by a circular walkway that is 4 meters wide. What is the area of the walkway?

To figure out the area of the walkway I found the difference between of the larger circle (circular lawn + circular walkway). To find the radius of the bigger circle I added the diameter of both the circular walkway and the circular lawn and divided by 2. (10+4)/2. I found the radius of the circle to be 7. So area of the bigger circle would be (7)squared times pi.

The Manhattan guide explains that the area of the bigger circle is 9(squared) times pie. Why is the radius of the bigger circle 9? I understand that the radius of the smaller circle is 5 but wouldn't the radius of the circular walkway be 2 since the entire walkway is 4 meters wide?

Jose

There is a figure attached to the original question. See below:
Attachment:
Circle.png
A circular lawn with a radius of 5 meters is surrounded by a circular walkway that is 4 meters wide (see figure). What is the area of the walkway?

Hope it heps.
Math Expert
Joined: 02 Sep 2009
Posts: 58418
Re: A circular lawn with a radius of 5 meters is surrounded by a circular  [#permalink]

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22 Jun 2013, 09:41
1
josemnz83 wrote:
Thanks Bunuel. I have each of the Manhattan guides so I had seen the figure before. I did not post it because I did not want to violate posting rules. Anyway even after looking at the figure I cannot understand why the radius of the walkway is not 2. Radius is half of the diameter. The problem says that the walkway is 4 meters wide. Is 4 meters not the diameter? Or is the radius 4 because the walkway is not really a "full circle" but more like a semicircle of sorts?

Thanks!

Bunuel wrote:
josemnz83 wrote:
A circular lawn with a radius of 5 meters is surrounded by a circular walkway that is 4 meters wide. What is the area of the walkway?

To figure out the area of the walkway I found the difference between of the larger circle (circular lawn + circular walkway). To find the radius of the bigger circle I added the diameter of both the circular walkway and the circular lawn and divided by 2. (10+4)/2. I found the radius of the circle to be 7. So area of the bigger circle would be (7)squared times pi.

The Manhattan guide explains that the area of the bigger circle is 9(squared) times pie. Why is the radius of the bigger circle 9? I understand that the radius of the smaller circle is 5 but wouldn't the radius of the circular walkway be 2 since the entire walkway is 4 meters wide?

Jose

There is a figure attached to the original question. See below:
Attachment:
The attachment Circle.png is no longer available
A circular lawn with a radius of 5 meters is surrounded by a circular walkway that is 4 meters wide (see figure). What is the area of the walkway?

Hope it heps.

Attachment:

Untitled.png [ 6.82 KiB | Viewed 8826 times ]

Circular lawn = green area;
Walkaway = yellow area.

Does this make sense?
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Re: A circular lawn with a radius of 5 meters is surrounded by a circular  [#permalink]

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22 Jun 2013, 09:48
josemnz83 wrote:

A circular lawn with a radius of 5 meters is surrounded by a circular walkway that is 4 meters wide. What is the area of the walkway?

To figure out the area of the walkway I found the difference between of the larger circle (circular lawn + circular walkway). To find the radius of the bigger circle I added the diameter of both the circular walkway and the circular lawn and divided by 2. (10+4)/2. I found the radius of the circle to be 7. So area of the bigger circle would be (7)squared times pi.

The Manhattan guide explains that the area of the bigger circle is 9(squared) times pie. Why is the radius of the bigger circle 9? I understand that the radius of the smaller circle is 5 but wouldn't the radius of the circular walkway be 2 since the entire walkway is 4 meters wide?

Jose

Area of smaller circle = 5pie

Now how would you calculate area of walkway= Area of smaller circle- area of bigger circle(includes walkway)
= 5- (5+4) = 5-9= 4
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Re: A circular lawn with a radius of 5 meters is surrounded by a circular  [#permalink]

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22 Jun 2013, 09:55
Thanks for the help! I feel really dumb right now. lol

What I failed to recognize is that the 4 was indicating half of the walkway outside of the circular lawn. The diameter of the whole circle was 18 (10 for the lawn + 4 for one "side" of the walkway" + 4 for other side). Radius then would be 9 pi. The rest I did not have trouble with.

Thanks so much for the prompt replies!

prateekbhatt wrote:
josemnz83 wrote:

A circular lawn with a radius of 5 meters is surrounded by a circular walkway that is 4 meters wide. What is the area of the walkway?

To figure out the area of the walkway I found the difference between of the larger circle (circular lawn + circular walkway). To find the radius of the bigger circle I added the diameter of both the circular walkway and the circular lawn and divided by 2. (10+4)/2. I found the radius of the circle to be 7. So area of the bigger circle would be (7)squared times pi.

The Manhattan guide explains that the area of the bigger circle is 9(squared) times pie. Why is the radius of the bigger circle 9? I understand that the radius of the smaller circle is 5 but wouldn't the radius of the circular walkway be 2 since the entire walkway is 4 meters wide?

Jose

Area of smaller circle = 5pie

Now how would you calculate area of walkway= Area of smaller circle- area of bigger circle(includes walkway)
= 5- (5+4) = 5-9= 4
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Re: A circular lawn with a radius of 5 meters is surrounded by a circular  [#permalink]

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10 Oct 2013, 20:37
2
Radius of larger circle (Walk area inclusive) = 5 +4 =9
Area = Pie (9) ^2 = 81pie......................... (1)

Radius of smaller circle = 5
Area = Pie (5) ^2 = 25pie.......................... (2)

Area of walkway = (1) - (2) = 56pie
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Re: A circular lawn with a radius of 5 meters is surrounded by a circular  [#permalink]

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18 Nov 2013, 19:11
$$pi(R^2-r^2)$$.
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Re: A circular lawn with a radius of 5 meters is surrounded by a circular  [#permalink]

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07 Oct 2018, 22:52
Area of the smaller circle = πr^2 = π x5^2 = 25π
Area of the larger circle = πr^2 = π x 9^2 = 81π
Difference(Area of the walkway) = 81π - 25π = 56π

Hence B
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Re: A circular lawn with a radius of 5 meters is surrounded by a circular  [#permalink]

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24 Feb 2019, 16:09
Hi All,

We're told that a circular lawn with a radius of 5 meters is surrounded by a circular walkway that is 4 meters wide. We're asked for the area of the walkway. This is an example of a 'punch out' question - we need to find the area of the ENTIRE shape and then 'punch out' the part that we don't want.

The radius of the ENTIRE circle is (5+4) = 9, so the area of the large circle is (R^2)π = (9^2)π = 81π
The radius of the lawn is 5, so the area of the small circle is (R^2)π = (5^2)π = 25π

The area of the walkway is the total area - the area of the law... 81π - 25π = 56π

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Re: A circular lawn with a radius of 5 meters is surrounded by a circular   [#permalink] 24 Feb 2019, 16:09
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