A sphere and a cylinder have the same radius and height. The volume of the cylinder is $18 cm^3$. What is the volume of the sphere?
A. $12 cm^3$
B. $24 cm^3$
C. $36 cm^3$
Answer (2)
Define the radius r and use the relationship h = 2 r between the cylinder's height and the sphere's radius.
Express the cylinder's volume as V cy l in d er = 2 π r 3 and set it equal to 18 c m 3 , then solve for π r 3 .
Use the sphere's volume formula V s p h ere = 3 4 π r 3 and substitute the value of π r 3 .
Calculate the sphere's volume: 12 c m 3 .
Explanation
Problem Analysis Let's analyze the problem. We are given a sphere and a cylinder with the same radius and height. We know the volume of the cylinder and need to find the volume of the sphere. The key is to relate the volumes using the given information.
Define variables and relationships Let r be the radius of both the sphere and the cylinder, and let h be the height of the cylinder. Since the sphere and cylinder have the same height, the height of the cylinder is equal to the diameter of the sphere, so h = 2 r .
Cylinder Volume The volume of the cylinder is given by the formula: V cy l in d er = π r 2 h Since h = 2 r , we can substitute this into the formula: V cy l in d er = π r 2 ( 2 r ) = 2 π r 3 We are given that the volume of the cylinder is 18 c m 3 , so: 2 π r 3 = 18
Sphere Volume The volume of the sphere is given by the formula: V s p h ere = 3 4 π r 3
Solve for π r 3 From the cylinder volume equation, we have 2 π r 3 = 18 . We can divide both sides by 2 to get: π r 3 = 9
Calculate Sphere Volume Now, substitute π r 3 = 9 into the sphere volume formula: V s p h ere = 3 4 π r 3 = 3 4 ( 9 ) V s p h ere = 3 4 × 9 = 3 36 = 12
Final Answer Therefore, the volume of the sphere is 12 c m 3 .
Examples
Understanding the relationship between the volumes of spheres and cylinders is useful in various real-world applications. For instance, when designing containers or tanks, engineers often need to calculate the volume of different shapes to optimize space and material usage. Knowing that a sphere's volume is related to a cylinder's volume (with equal radius and height) by a factor of 3 2 allows for quick estimations and comparisons in design scenarios. This can help in determining the most efficient shape for storing liquids or gases, minimizing waste, and reducing costs.
The volume of the sphere is calculated to be 12 cm³ using the relationship between the volumes of a cylinder and a sphere with the same radius and height. Therefore, the correct choice is A. 12 cm³.
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