A cone has a volume of 5 cubic inches. What is the volume of a cylinder that the cone fits exactly inside of?
A. $15 in ^3$
B. $20 in ^3$
C. $25 in ^3$
D. $30 in ^3$
Answer (1)
The volume of a cone is 3 1 π r 2 h .
The volume of a cylinder is π r 2 h .
Since the cone fits exactly inside the cylinder, the volume of the cylinder is three times the volume of the cone.
Given the cone's volume is 5 cubic inches, the cylinder's volume is 15 i n 3 .
Explanation
Problem Analysis We are given that a cone has a volume of 5 cubic inches and we need to find the volume of a cylinder that the cone fits exactly inside.
Volume Formulas The volume of a cone is given by the formula: V co n e = 3 1 π r 2 h where r is the radius of the base and h is the height of the cone. 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.
Relating Cone and Cylinder Volumes Since the cone fits exactly inside the cylinder, they share the same radius r and height h . Therefore, we can express the volume of the cylinder in terms of the volume of the cone: V cy l in d er = 3 × 3 1 π r 2 h = 3 V co n e We are given that V co n e = 5 cubic inches.
Calculating Cylinder Volume Substituting the given volume of the cone into the equation: V cy l in d er = 3 × 5 = 15 Therefore, the volume of the cylinder is 15 cubic inches.
Final Answer The volume of the cylinder is 15 cubic inches.
Examples
Imagine you're designing containers for ice cream. You know a cone-shaped container holds 5 cubic inches of ice cream. If you want to design a cylindrical container that perfectly fits the same cone inside, you'll need a cylinder with a volume three times larger, which is 15 cubic inches. This ensures no ice cream is wasted and the cone fits snugly within the cylinder.
