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An ice cream cone consists of a sphere of vanilla ice cream.

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An ice cream cone consists of a sphere of vanilla ice cream.

by pzazz12 » Fri Oct 08, 2010 5:17 am
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. Asslime 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?

A. 2:1
B. 3:1
C. 4:1
D. 16:3
E. 6:1
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by euro » Fri Oct 08, 2010 5:35 am
Volume of frozen ice-cream = Volume of sphere = (4/3) Pi m^3 where m is the radius of sphere

Volume of melted ice-cream
= 75% of frozen
= (3/4) * (4/3) Pi m^3
= Pi m^3

Volume of cone
= (1/3) *Pi *r^2 * h where r = radius and h = height of cone

Volume of cone = Volume of melted ice-cream

Pi * m^3 = (1/3) *Pi *r^2 * h

or m^3 = (1/3) *r^2 * h

Its given in the problem that m = r, thus we have

=> r^3 = (1/3) *r^2 * h

=> r = (1/3) * h

[spoiler] => h/r = 3/1 - (B)[/spoiler] Answer
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by Rahul@gurome » Fri Oct 08, 2010 5:35 am
Let the radius of cone = sphere of cone = r and h be the height of cone.
Volume of frozen ice cream = (4/3)(pi)(r^3)
Volume of melted ice cream = 0.75 * (4/3)(pi)(r^3) = (pi)(r^3)
Volume of cone = (1/3)(pi)(r^2)h
Since, when the ice cream melts, it will exactly fill the cone, so volume of melted ice cream = Volume of cone
(pi)(r^3) = (1/3)(pi)(r^2)h implies r = h/3
h: r = 3:1

[spoiler]The correct answer is (B).[/spoiler]
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by pzazz12 » Fri Oct 08, 2010 11:25 pm
Rahul@gurome wrote:Let the radius of cone = sphere of cone = r and h be the height of cone.
Volume of frozen ice cream = (4/3)(pi)(r^3)
Volume of melted ice cream = 0.75 * (4/3)(pi)(r^3) = (pi)(r^3)
Volume of cone = (1/3)(pi)(r^2)h
Since, when the ice cream melts, it will exactly fill the cone, so volume of melted ice cream = Volume of cone
(pi)(r^3) = (1/3)(pi)(r^2)h implies r = h/3
h: r = 3:1

[spoiler]The correct answer is (B).[/spoiler]
thank you....know i understood how to solve...but is there any shortcut approach to solve it...
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