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| The ice cream cone (Posted on 2010-03-01) |
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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?
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Submitted by Dej Mar
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Rating: 5.0000 (1 votes)
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Solution:
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(Hide)
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26 cents.
The volume of a sphere is (4/3)*PI*r3. With a diameter of 4 inches, the radius of the sphere is 2 inches and the volume is then (32/3)*PI cubic-inches.
The volume of a cone is given as (h/3)*PI*r2 where h is the height of the cone and r is the radius of the base of the cone. With the cone's height of 10 inches and the cone's spatial volume being equal to the volume of the the scoop of ice cream, the equation can be simplified and expressed as r = 4*SQRT(1/5).
The base of the cone is shared with the base of the spherical cap, with the spherical cap being the bounded volume of the scoop of ice cream below the rim of the cone. The height of spherical cap is equivalent to the length of the sagitta. The length of the sagitta may be calculated with the Pythagorean equation: (r - s)2 + a2 = r2, where r is the radial length of the sphere, s is the length of the sagitta, and a is the radial length of the base of the spherical cap. Using in the value of the sphere's radius, 2, and the radial length of the base of the spherical cap, 4*SQRT(1/5), a quadratic equation may be expressed: s2 - 4s + 16/5 = 0. Solving for the roots of the quadratic, the roots are found to be s1 = 2+2*SQRT(1/5) and s2 = 2-2*SQRT(1/5). As the sagitta is less than the radius of the sphere, the value for s is the root 2-2*SQRT(1/5).
The equation for the volume of a spherical cap is (1/6)*PI*h*(3a2 + h2) where h is the height of the cap and a is the base radius of the cap. Given the base radius of the cap, 4*SQRT(1/5), and the height of the cap, 2-2*SQRT(1/5), the volume is calculated to be (80 - 112*SQRT(1/5))*(PI/15) cubic-inches.
The percentage of the ice cream above the base of the cone is 100*(Volumesphere - Volumecap)/Volumesphere, which is calculated to be approximately 81.30495%. Rounded to the nearest integer, 81%. An 81% discount to the $1 price is 19 cents. |
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