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# P and Q are each circular regions. What is the radius of Q, if the are

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Math Expert
Joined: 02 Sep 2009
Posts: 64173
P and Q are each circular regions. What is the radius of Q, if the are  [#permalink]

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11 Sep 2018, 00:54
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25% (medium)

Question Stats:

81% (01:34) correct 19% (03:25) wrong based on 37 sessions

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P and Q are each circular regions. What is the radius of Q, if the area of P minus the area of Q is 15π and P has a circumference that is 4 times that of Q?

A. 1/4
B. 1
C. 2
D. 4
E. 20

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Posts: 364
P and Q are each circular regions. What is the radius of Q, if the are  [#permalink]

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11 Sep 2018, 03:02
Bunuel wrote:
P and Q are each circular regions. What is the radius of Q, if the area of P minus the area of Q is 15π and P has a circumference that is 4 times that of Q?

Let the radius of Circular region P be p and that of Q be q
Given:
$$2\pi*p=4*2\pi*q$$
this implies, $$p=4q$$.................a

Also, $$\pi*p^2-\pi*q^2=15\pi$$
therefore, $$p^2-q^2=15$$
Using a, we get
$$16q^2-q^2=15$$
$$15q^2=15$$
therefore, $$q^2=1$$or $$q=1$$

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Re: P and Q are each circular regions. What is the radius of Q, if the are  [#permalink]

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14 Sep 2018, 16:53
Bunuel wrote:
P and Q are each circular regions. What is the radius of Q, if the area of P minus the area of Q is 15π and P has a circumference that is 4 times that of Q?

A. 1/4
B. 1
C. 2
D. 4
E. 20

Let r = the radius of circle Q and R = the radius of circle P. We have

πR^2 - πr^2 = 15π

and

2πR = 4(2πr)

Simplifying the equations, we have R^2 - r^2 = 15 and R = 4r, respectively. Substituting 4r into R in R^2 - r^2 =15, we have:

(4r)^2 - r^2 = 15

16r^2 - r^2 = 15

15r^2 = 15

r^2 = 1

r = 1

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# Scott Woodbury-Stewart

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Joined: 15 Sep 2018
Posts: 31
Re: P and Q are each circular regions. What is the radius of Q, if the are  [#permalink]

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17 Sep 2018, 12:13
Let:
p be the radius of circle P
q be the radius of circle Q

Take note that the formula for circumference is:

Find the circumference of each circle:
Circumference of P = 2πp
Circumference of Q = 2πq

We can use the relationship between the circumferences of circle P and Q to determine the relationship between their radii.

Circumference of P = 4 × Circumference of Q
2πp = 4 × 2πq
2πp = 8πq
(2πp/2π) = (8πp/2π)

p = 4q

Take note that the formula for area is:

Find the area of each circle:
Area of P = πp^2
Area of Q = πq^2

Use the relationship of the areas of circle P and Q to determine the exact value of their radii.
Area of P – Area of Q = 15π
πp^2 – πq^2 = 15π
Take note that p = 4q
π(4q)^2 – πq^2 = 15π
16πq^2 – πq^2 = 15π
15πq^2 = 15π

Find the value of q:
15πq^2/15π = 15π/15π

q^2 = 1

√q^2 = √1
q = 1

Therefore, the radius of Q is 1.

B) 1
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Re: P and Q are each circular regions. What is the radius of Q, if the are  [#permalink]

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18 Sep 2018, 17:50
Bunuel wrote:
P and Q are each circular regions. What is the radius of Q, if the area of P minus the area of Q is 15π and P has a circumference that is 4 times that of Q?

A. 1/4
B. 1
C. 2
D. 4
E. 20

We let the radius of P = P and the radius of Q = Q and create the equations:

πP^2 - πQ^2 = 15π

P^2 - Q^2 = 15

and

2Pπ = 4(2Qπ)

Pπ = 4Qπ

P = 4Q

Substituting, we have:

(4Q)^2 - Q^2 = 15

16Q^2 - Q^2 = 15

15Q^2 = 15

Q = 1

_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
202 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Re: P and Q are each circular regions. What is the radius of Q, if the are   [#permalink] 18 Sep 2018, 17:50