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An ice cream cone consists of a sphere of vanilla ice cream and a righ

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An ice cream cone consists of a sphere of vanilla ice cream and a righ  [#permalink]

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New post 20 Mar 2019, 23:07
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A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

86% (03:25) correct 14% (05:40) wrong based on 7 sessions

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An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies 75% of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius? (cone volume \(=\frac{πr^2h}{3}\); sphere volume \(= \frac{4πr^3}{3}\))

(A) 2:1
(B) 3:1
(C) 4:1
(D) 16:3
(E) 6:1

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Re: An ice cream cone consists of a sphere of vanilla ice cream and a righ  [#permalink]

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New post 20 Mar 2019, 23:23

Solution


Given:
    • An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere.
    • If the ice cream melts, it will exactly fill the cone.
    • The melted ice cream occupies 75% of the volume of the frozen ice cream.

To find:
    • The ratio of the cone’s height to its radius.

Approach and Working:
As the diameter of cone and sphere is same, let us assume that their radius is r.
    • The maximum ice-cream the cone can hold = the volume of cone = \(\frac{πr^2h}{3}\)
    • Volume of frozen ice cream = volume of sphere = \(\frac{4πr^3}{3}\)
    • Volume of melted ice cream = 75% of volume of frozen ice cream = \(\frac{3}{4} * \frac{4πr^3}{3} = πr^3\)

As per the given condition,
    • \(πr^3 = \frac{πr^2h}{3}\)
    Or, r = h/3
    Or, h: r = 3: 1

Hence, the correct answer is option B.

Answer: B

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Re: An ice cream cone consists of a sphere of vanilla ice cream and a righ  [#permalink]

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New post 21 Mar 2019, 00:31
Bunuel wrote:
An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies 75% of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius? (cone volume \(=\frac{πr^2h}{3}\); sphere volume \(= \frac{4πr^3}{3}\))

(A) 2:1
(B) 3:1
(C) 4:1
(D) 16:3
(E) 6:1


vol of sphere = \(= \frac{4πr^3}{3}\))

so vol of cone = .75 * \(= \frac{4πr^3}{3}\)) = \(=\frac{πr^2h}{3}\)
solve we get
h/r = 3:1
IMO B
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Re: An ice cream cone consists of a sphere of vanilla ice cream and a righ   [#permalink] 21 Mar 2019, 00:31
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