A solid metal sculpture in Kamila’s backyard is in the shape of a sphere and has a radius of 6 inches. She wants to have it melted and made into small solid cones. She would like the cones to each have a diameter of 6 inches and a height of 9 inches. With these dimensions, how many cones can be made from the metal sphere?In your final answer, include all formulas, equations, and calculations necessary to determine how many cones can be made.
Question
A solid metal sculpture in Kamila’s backyard is in the shape of a sphere and has a radius of 6 inches. She wants to have it melted and made into small solid cones. She would like the cones to each have a diameter of 6 inches and a height of 9 inches. With these dimensions, how many cones can be made from the metal sphere?In your final answer, include all formulas, equations, and calculations necessary to determine how many cones can be made.
Solution
To solve this problem, we need to find the volume of the sphere and the volume of the cone, then divide the volume of the sphere by the volume of the cone to find out how many cones can be made from the sphere.
- Find the volume of the sphere:
The formula for the volume of a sphere is (4/3)πr³. The radius of the sphere is given as 6 inches.
V_sphere = (4/3)π(6)³ = 288π cubic inches.
- Find the volume of the cone:
The formula for the volume of a cone is (1/3)πr²h. The radius of the cone is half the diameter, so r = 3 inches. The height is given as 9 inches.
V_cone = (1/3)π(3)²(9) = 27π cubic inches.
- Divide the volume of the sphere by the volume of the cone to find out how many cones can be made:
Number of cones = V_sphere / V_cone = 288π / 27π = 10.67
Since we can't have a fraction of a cone, we round down to the nearest whole number.
So, Kamila can make 10 cones from the metal sphere.
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