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 The ice cream cone (Posted on 2010-03-01)
David recently visited Pop's Ice Cream Shoppe and ordered the \$1 sugar cone with one scoop of vanilla ice cream placed firmly atop the cone.

Pop told David that he would receive a percentage discount in the cost of the treat equal to the closest integer value to the answer of the following question if he answered it correctly:

"If the sugar cone is a right circular cone with a height of 10 inches, and the scoop of vanilla ice cream is a perfect sphere with a diameter of 4 inches, and both the cone and sphere are equal in spatial volume, what percentage of ice cream is above the base of the cone?"

David, a bright student, gave a correct answer. How much did David pay for the ice cream cone?

 See The Solution Submitted by Dej Mar Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Solution | Comment 1 of 2
`Let H be the height of the cone and rthe radius of its base. Let R be theradius of the scoop of ice cream andp the height of its center above thecone. `
`If you cut a sphere, of radius R, witha plane, that is a distance b from thecenter of the sphere, then the volume of the largest portion is`
`  (pi/3)*[2*a^3 + b*(3*a^2 - b^2)]  (1)`
`The volume of the sphere and the coneare equal, so`
`  4*(pi/3)*R^3 = (pi/3)*r^2*H`
`               or`
`           r^2 = 4*R^3/H            (2)`
`The relation of R, p, and r is   `
`     p^2 + r^2 = R^2                (3)`
`Combining (2) and (3) we get`
`  p = R*sqrt(1 - 4*R/H)`
`Plugging R and p into (1) for a and bwe get the amount of ice cream above thecone`
`  (pi/3)*R^3*[2 + sqrt(1 - 4*R/H)*(2 + 4*R/H)]`
`Multiplying by 100 and dividing by thevolume of the sphere we get the percent`
`  25*[2 + sqrt(1 - 4*R/H)*(2 + 4*R/H)]`
`For our problem with H = 10 and R = 2,`
`  50 + 14*sqrt(5) =~ 81.30495.`
` `

 Posted by Bractals on 2010-03-01 16:14:57

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