A sphere and a cylinder have the same radius and height. The volume of the cylinder is $54 m^3$. Amie found the volume of the sphere.
Her work is shown below.
$V=\frac{2}{3}+54$
Answer (2)
The volume of the cylinder is given as 54 m 3 , and the height of the cylinder is equal to the diameter of the sphere ( h = 2 r ).
Express the volume of the cylinder in terms of its radius r : V cy l in d er = π r 2 h = 2 π r 3 = 54 m 3 .
Calculate r 3 = π 27 .
Calculate the volume of the sphere using the formula V s p h ere = 3 4 π r 3 = 3 4 π ⋅ π 27 = 36 m 3 . The volume of the sphere is 36 m 3 .
Explanation
Problem Analysis We are given that a sphere and a cylinder have the same radius, r , and height, h . The volume of the cylinder is 54 m 3 . We need to find the volume of the sphere.
Cylinder Volume The volume of a cylinder is given by the formula: V cy l in d er = π r 2 h
Height and Radius Since the sphere and cylinder have the same radius and height, the height of the cylinder is equal to the diameter of the sphere. Therefore, we have: h = 2 r
Substitute Height Substitute h = 2 r into the volume of the cylinder equation: π r 2 ( 2 r ) = 2 π r 3 = 54 m 3
Solve for r^3 Solve for r 3 :
r 3 = 2 π 54 = π 27 m 3
Sphere Volume The volume of a sphere is given by the formula: V s p h ere = 3 4 π r 3
Calculate Sphere Volume Substitute the value of r 3 into the volume of the sphere equation: V s p h ere = 3 4 π ( π 27 ) = 3 4 ( 27 ) = 4 ( 9 ) = 36 m 3
Final Answer Therefore, the volume of the sphere is 36 m 3 . Amie's work is incorrect.
Examples
Understanding the relationship between the volumes of spheres and cylinders is useful in various real-world applications. For example, in manufacturing, if you know the volume of a cylindrical container, you can determine the volume of a spherical ball that would fit inside it, or vice versa. This is also applicable in fields like architecture, where you might need to calculate the volume of spherical domes or cylindrical pillars.
The volume of the sphere, given that its radius and height are the same as a cylinder with a volume of 54 m³, is calculated to be 36 m³. Amie's calculation was incorrect. The correct method gives us the volume of the sphere as 36 m³.
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