The formula for the volume of a cone is V = 1/3πr^2h, where r is the radius of the cone's base and h is the height of the cone.
The formula for the volume of a sphere is V = 4/3πr^3, where r is the radius of the sphere.
Since the diameter of the ice cream cone is equal to its height, let's call that value x. The radius of the base of the cone is then x/2.
The volume of the cone is given by V = 1/3π(x/2)^2(x) = 1/3π(x^3/4).
The volume of the hemisphere on top of the cone is given by V = 4/3π(r^3/2) = 2/3π(x^3/8).
The total volume of the ice cream is the sum of the volume of the cone and the volume of the hemisphere: (1/3π(x^3/4)) + (2/3π(x^3/8)) = (3x^3 + 2x^3)π/12 = 5x^3π/12.
We are given that the volume of the hemisphere is 4 in^3, so 2/3π(x^3/8) = 4. Simplifying this equation, we get x^3/12 = 4/π. Multiplying both sides by 12/π, we get x^3 = 48/π. Taking the cube root of both sides, we get x = (48/π)^(1/3).
Substituting this value of x into the formula for the total volume of the ice cream, we get (5((48/π)^(1/3))^3π)/12 = (5(48/π))/12 = (20/π) in^3.
So the total volume of ice cream in and on top of the cone is (20/π) in^3, or approximately 6.366 in^3.
Aimee packs ice cream into an ice cream cone. She then puts a perfect hemisphere of ice cream on top of the cone that has a volume of 4 in.^3 The diameter of the ice cream cone is equal to its height. What is the total volume of ice cream in and on top of the cone? Use the relationship between the formulas for the volumes of cones and spheres to help solve this problem. Show your work and explain your reasoning.
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just give me the answer and make sure its correct. MAKE SURE!!
The correct answer is (20/π) in^3, or approximately 6.366 in^3.
1 answer