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Intern  Joined: 18 Mar 2012
Posts: 46
GPA: 3.7
The area of Circle O is added to its diameter. If the circum  [#permalink]

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Question Stats: 85% (02:07) correct 15% (03:13) wrong based on 107 sessions

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The area of Circle O is added to its diameter. If the circumference of Circle O is then subtracted from this total, the result is 4. What is the radius of Circle O?

A. 2/π
B. 2
C. 3
D. 4
E. 5
Math Expert V
Joined: 02 Sep 2009
Posts: 58402
Re: The area of Circle O is added to its diameter. If the circum  [#permalink]

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alexpavlos wrote:
The area of Circle O is added to its diameter. If the circumference of Circle O is then subtracted from this total, the result is 4. What is the radius of Circle O?

A. 2/π
B. 2
C. 3
D. 4
E. 5

Given: area+diameter-circumference=4 --> $$\pi{r^2}+2r-2\pi{r}=4$$ --> $$(\pi{r^2}-2\pi{r})+(2r-4)=$$ --> $$\pi{r}({r-2)+2(r-2)=0$$ --> $$(r-2)(\pi{r}+2)=0$$ --> either $$r=2$$ or $$r=-\frac{2}{\pi}$$, which is not a valid solution since the value of the radius cannot be negative.

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The area of Circle O is added to its diameter. If the circum  [#permalink]

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Area + Diameter - Circumference = 4

Just observe the above setup equation in terms of circle

Area calculation always involves the $$\pi$$ factor. Example, $$\pi r^2$$ etc for whatever the radius may be.

Similarly, Circumference calculation always involves again the $$\pi$$ factor. Example, $$2\pi r$$ for whatever the radius may be.

Now the resultant of all this ideally should have a $$\pi$$. However , the resultant returned = 4.

It means the $$\pi$$ factor is cancelled out in the calculation which also means Area of the given circle = circumference of the given circle.

Area of Circle = Circumference of circle: Happens only when Radius = 2

In this case, Diameter = 4; so $$radius = \frac{4}{2} = 2$$

Bunuel/Experts: Kindly let me know if above argument holds value for calculation purpose in this question? Thank you
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Re: The area of Circle O is added to its diameter. If the circum  [#permalink]

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alex1233 wrote:
The area of Circle O is added to its diameter. If the circumference of Circle O is then subtracted from this total, the result is 4. What is the radius of Circle O?

A. 2/π
B. 2
C. 3
D. 4
E. 5

Bunuel/ Karishma: Although I have posted my answer earlier, want to know if this problem is "technically correct"?

Say, if the diameter is in meters, then circumference would also be in meters. However, Area would be in $$(meter)^2$$.

How can we perform addition/subtraction in dissimilar units?
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Re: The area of Circle O is added to its diameter. If the circum  [#permalink]

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PareshGmat wrote:
alex1233 wrote:
The area of Circle O is added to its diameter. If the circumference of Circle O is then subtracted from this total, the result is 4. What is the radius of Circle O?

A. 2/π
B. 2
C. 3
D. 4
E. 5

Bunuel/ Karishma: Although I have posted my answer earlier, want to know if this problem is "technically correct"?

Say, if the diameter is in meters, then circumference would also be in meters. However, Area would be in $$(meter)^2$$.

How can we perform addition/subtraction in dissimilar units?

I think it's clear that the question means adding numerical value of area to numerical value of diameter. I see no point to complicate.
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Re: The area of Circle O is added to its diameter. If the circum  [#permalink]

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area+diameter-circumference=4

pi*r^2+2r=4+2pi*r

isolate r and get r(pi*r+2)=4+2pi*r

r=(4+2pi*r)/(pi*r+2) =>2(2+pi*r)/(pi*r+2)

r=2

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Posts: 4007
Re: The area of Circle O is added to its diameter. If the circum  [#permalink]

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Top Contributor
alex1233 wrote:
The area of Circle O is added to its diameter. If the circumference of Circle O is then subtracted from this total, the result is 4. What is the radius of Circle O?

A. 2/π
B. 2
C. 3
D. 4
E. 5

Area = πr²
Diameter = 2r

Circumference = 2πr

We can write: πr² + 2r - 2πr = 4
Rearrange and set equal to zero to get: πr² - 2πr + 2r - 4 = 0
Factor IN PARTS: πr(r - 2) + 2(r - 2) = 0
Combine to get: (πr + 2)(r - 2) = 0

So, EITHER πr + 2 = 0 OR r - 2 = 0
If πr + 2 = 0, then r = -2/π (not possible)
If r - 2 = 0, then r = 2 (perfect!)

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_________________ Re: The area of Circle O is added to its diameter. If the circum   [#permalink] 11 Jan 2018, 08:21
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