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

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Math Expert
Joined: 02 Sep 2009
Posts: 59725
An ice cream cone consists of a sphere of vanilla ice cream and a righ  [#permalink]

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20 Mar 2019, 23:07
00:00

Difficulty:

35% (medium)

Question Stats:

69% (03:04) correct 31% (02:58) wrong based on 13 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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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.

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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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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
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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