The volume of a cylinder is $156 \pi \text{ cm}^3$. What is the volume of a cone with the same base and height as the cylinder?
A. $39 \pi \text{ cm}^3$
B. $52 \pi \text{ cm}^3$
C. $78 \pi \text{ cm}^3$
D. $156 \pi \text{ cm}^3$
Answer (2)
The problem states that a cylinder has a volume of 156 π c m 3 and asks for the volume of a cone with the same base and height.
Recall the formulas for the volume of a cylinder, V cy l in d er = π r 2 h , and the volume of a cone, V co n e = 3 1 π r 2 h .
Recognize that the volume of the cone is 3 1 the volume of the cylinder, so V co n e = 3 1 V cy l in d er .
Calculate the volume of the cone: V co n e = 3 1 ( 156 π ) = 52 π c m 3 . The final answer is 52 π c m 3 .
Explanation
Problem Analysis We are given that the volume of a cylinder is 156 c m 3 . We need to find the volume of a cone that has the same base and height as the cylinder.
Volume Formulas The volume of a cylinder is given by the formula: V cy l in d er = π r 2 h where r is the radius of the base and h is the height of the cylinder.
The volume of a cone with the same base and height is given by the formula: V co n e = 3 1 π r 2 h Notice that the volume of the cone is one-third the volume of the cylinder, given that they have the same base and height.
Calculating the Cone Volume Since the cylinder and cone have the same base and height, we can write: V co n e = 3 1 V cy l in d er We are given that V cy l in d er = 156 π c m 3 . Substituting this value into the equation, we get: V co n e = 3 1 ( 156 π ) c m 3 V co n e = 52 π c m 3
Final Answer Therefore, the volume of the cone is 52 π c m 3 .
Examples
Understanding the relationship between the volumes of cylinders and cones is useful in various real-world applications. For example, when designing containers or structures, knowing that a cone's volume is one-third of a cylinder with the same base and height helps in estimating material requirements and optimizing space. This principle is applied in architecture, engineering, and even in culinary arts when dealing with conical or cylindrical shapes.
The volume of a cone with the same base and height as a cylinder with a volume of 156 π cm 3 is 52 π cm 3 , which is one-third of the cylinder's volume. Thus, the correct option is B. 52 π cm 3 .
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