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

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

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New post 11 Sep 2018, 01:54
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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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P and Q are each circular regions. What is the radius of Q, if the are  [#permalink]

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New post 11 Sep 2018, 04:02
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 \(q=1\)

Answer: B
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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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New post 14 Sep 2018, 17:53
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

Answer: B
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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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New post 17 Sep 2018, 13:13
Let:
p be the radius of circle P
q be the radius of circle Q

Take note that the formula for circumference is:
C = 2π × radius

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:
A = π × (radius)2

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.

Answer:
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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New post 18 Sep 2018, 18:50
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

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