A sphere and a cylinder have the same radius and height. The volume of the cylinder is $30 m^3$. What is the volume of the sphere?
Answer (2)
Express the volume of the cylinder as V cy l in d er = π r 2 h and use the fact that h = 2 r to get V cy l in d er = 2 π r 3 .
Use the given volume of the cylinder, 30 m 3 , to find that 2 π r 3 = 30 , which simplifies to π r 3 = 15 .
Express the volume of the sphere as V s p h ere = 3 4 π r 3 .
Substitute the value of π r 3 into the sphere volume formula to find the volume of the sphere: V s p h ere = 3 4 ( 15 ) = 20 m 3 . The volume of the sphere is 20 m 3 .
Explanation
Problem Analysis Let's analyze the problem. We are given a sphere and a cylinder with the same radius, r , and height, h . We also know that the volume of the cylinder is 30 m 3 . Our goal is to find the volume of the sphere.
Volume of Cylinder First, let's write down the formula for the volume of a cylinder: V cy l in d er = π r 2 h
Substitute h = 2r Since the height of the cylinder is the same as the diameter of the sphere, we have h = 2 r . Substituting this into the volume formula for the cylinder, we get: V cy l in d er = π r 2 ( 2 r ) = 2 π r 3
Use Given Volume We are given that the volume of the cylinder is 30 m 3 , so we can write: 2 π r 3 = 30
Volume of Sphere Now, let's find the formula for the volume of a sphere: V s p h ere = 3 4 π r 3
Solve for pi*r^3 From the cylinder's volume equation, we can express π r 3 as: π r 3 = 2 30 = 15
Substitute into Sphere Volume Now, substitute this value into the sphere's volume equation: V s p h ere = 3 4 ( 15 ) = 20
Final Answer Therefore, the volume of the sphere is 20 m 3 .
Examples
Understanding the volumes of spheres and cylinders is useful in many real-world applications. For example, when designing storage tanks for liquids or gases, engineers need to calculate the volumes accurately to ensure the tanks meet the required capacity. Also, in architecture, knowing the relationship between the volumes of different shapes helps in creating aesthetically pleasing and structurally sound designs. Consider a spherical water tank and a cylindrical support structure; calculating their volumes ensures the structure can adequately support the tank's weight and volume.
The volume of the sphere, which has the same radius as the cylinder, is calculated to be 20 m³ using the provided volume of the cylinder. This was achieved by substituting the radius and height relationship into the volume formula for both shapes. Ultimately, the calculated volume of the sphere is 20 m³.
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