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If the radius of a circle were increased by 5 feet, the area of the

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If the radius of a circle were increased by 5 feet, the area of the circular region would increase by 65π square feet. What is the radius, in feet, before the increase?

(A) 4
(B) 8
(C) 9
(D) 13
(E) 18
[Reveal] Spoiler: OA

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Re: If the radius of a circle were increased by 5 feet, the area of the [#permalink]

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Bunuel wrote:
If the radius of a circle were increased by 5 feet, the area of the circular region would increase by 65π square feet. What is the radius, in feet, before the increase?

(A) 4
(B) 8
(C) 9
(D) 13
(E) 18


Let r = radius of ORIGINAL circle
Area = π(radius)²
So, area of ORIGINAL circle = πr²

If the radius of a circle were increased by 5 feet....
So, r + 5 = radius of NEW circle
Area of NEW circle = π(r + 5)²

...the area of the circular region would increase by 65π square feet.
(area of NEW circle) = (area of ORIGINAL circle) + 65π
In other words, π(r + 5)² = πr² + 65π
Divide both sides by π to get: (r + 5)² = r² + 65
Expand and simplify left side: r² + 10r + 25 = r² + 65
Subtract r² from both sides to get: 10r + 25 = 65
Subtract 25 from both sides to get: 10r = 40
Solve: r = 4

What is the radius, in feet, before the increase?
Answer = 4
= A

Cheers,
Brent
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If the radius of a circle were increased by 5 feet, the area of the [#permalink]

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New post 22 Nov 2017, 08:15
Bunuel wrote:
If the radius of a circle were increased by 5 feet, the area of the circular region would increase by 65π square feet. What is the radius, in feet, before the increase?

(A) 4
(B) 8
(C) 9
(D) 13
(E) 18

Another way: use the answer choices.

Start with (C)
\(r = 9\)
Original area: \(πr^2 = 81π\)

New radius: \((9 + 5) = 14\)
New area: \(196π\)

Difference: \((196 - 81)π = 115π\)

That difference is too great by a lot -- almost twice 65π. Eliminate Answers D and E (the gap between difference of squares increases as numbers get larger, and we're adding the same constant).

Answer B, r = 8, is too close to 9. Try Answer (A)

Original \(r = 4\)
Original area = \(16π\)

New radius: \((4 + 5) = 9\)
New area: \(81π\)

Difference: \((81π - 16π) = 65π\)
That's a match.

Answer A
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Re: If the radius of a circle were increased by 5 feet, the area of the [#permalink]

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New post 22 Nov 2017, 23:13
Hey guys what is the OA? Is it A?
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Re: If the radius of a circle were increased by 5 feet, the area of the   [#permalink] 22 Nov 2017, 23:32
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