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The volume of a cylindrical tank is directly proportional to the heigh

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The volume of a cylindrical tank is directly proportional to the heigh [#permalink]

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The volume of a cylindrical tank is directly proportional to the height and the square of the radius of the tank. If a certain tank with a radius of 10 centimeters has a volume of 20,000 cubic centimeters, what is the volume, in cubic centimeters, of a tank of the same height with a radius of 15 centimeters?

(A) 300,000
(B) 45,000
(C) 30,000
(D) 15,000
(E) 4,500
[Reveal] Spoiler: OA

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Re: The volume of a cylindrical tank is directly proportional to the heigh [#permalink]

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New post 10 Dec 2017, 04:46
Bunuel wrote:
The volume of a cylindrical tank is directly proportional to the height and the square of the radius of the tank. If a certain tank with a radius of 10 centimeters has a volume of 20,000 cubic centimeters, what is the volume, in cubic centimeters, of a tank of the same height with a radius of 15 centimeters?

(A) 300,000
(B) 45,000
(C) 30,000
(D) 15,000
(E) 4,500


\(\frac{10cm}{20,000cm^3}=\frac{15cm}{x}, 10cm*x=20,000cm^3*15, x=30,000cm^3\)

(C) is the answer.
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The volume of a cylindrical tank is directly proportional to the heigh [#permalink]

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New post 10 Dec 2017, 05:06
V ∝ h*r^2 , where v= volume, h= height and r= radius of a cylinder .
So V= (hr^2 )k where k= constant .

So V1= 20000 cm^3= {h(10 cm)^2}k
V2= {h(15 cm)^2 }k

Dividing V1 by V2 we get,

V1/V2= (10/15)^2,
20000/V2= (2/3)^2=4/9,
So V2= 45000 cm^3,

(B) is the answer .



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The volume of a cylindrical tank is directly proportional to the heigh [#permalink]

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New post 10 Dec 2017, 11:44
Bunuel wrote:
The volume of a cylindrical tank is directly proportional to the height and the square of the radius of the tank. If a certain tank with a radius of 10 centimeters has a volume of 20,000 cubic centimeters, what is the volume, in cubic centimeters, of a tank of the same height with a radius of 15 centimeters?

(A) 300,000
(B) 45,000
(C) 30,000
(D) 15,000
(E) 4,500

All measures are in centimeters; units omitted.

Direct proportion: When \(y\) increases, \(x\) increases

\(y = kx\)*, and \(\frac{y}{x}=k\)
\(k\) is the constant of proportionality: it does not change

Find \(k\) from the first scenario, and use \(k\) to find volume in the second scenario

Scenario 1 - find \(k\)
The volume of a cylindrical tank is directly proportional to the height and the square of the radius of the tank

\(V = (k)(h)(r^2)\) , and

\(\frac{V}{(h)(r^2)} = k\)

Height does not change. Remove it.
\(V_1 = 20,000\)
\(r_1 = 10\)

\(\frac{V_1}{(r_1)^2}= k\)

\(\frac{20,000}{(100)} = 200 = k\)

Scenario 2 - Use the same formula; and \(k\); and the given new radius to find new volume, \(V_2\)

\(k = 200\)
\(r_2 = 15\)
New volume, \(V_2\)?

\(\frac{V_2}{(r_2)^2} = k\)

\(\frac{V_2}{225} = 200\)

\(V_2 = (225)(200) = 45,000\)

Answer B

*Sometimes written \(y ∝ x\)
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The volume of a cylindrical tank is directly proportional to the heigh   [#permalink] 10 Dec 2017, 11:44
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