Understanding the Volume Relationship Between Two Cylindrical Containers

When dealing with objects of similar shape but different dimensions, it's important to understand how these dimensions affect each other. In this article, we will explore the mathematical relationship between the volume of a plastic cylinder and a glass cylinder, given their specific dimensions.

The Volume of the Plastic Cylinder

The volume of the plastic cylinder is given as ( V_{text{plastic}} 64 ) cubic units. The volume of a cylinder is calculated using the formula:

where ( r ) is the radius and ( h ) is the height of the cylinder. In this case, let ( r_p ) and ( h ) be the radius and height of the plastic cylinder, respectively. We can express this as:

( V_{text{plastic}} pi r_p^2 h 64 )

The Dimensions of the Glass Cylinder

The glass cylinder shares the same height ( h ) as the plastic cylinder but has a radius that is half the radius of the plastic cylinder:

( r_g frac{1}{2} r_p )

Calculating the Volume of the Glass Cylinder

Using the dimensions of the glass cylinder, we can calculate its volume using the same formula for the volume of a cylinder:

( V_{text{glass}} pi r_g^2 h )

Substituting ( r_g frac{1}{2} r_p ) into the equation:

( V_{text{glass}} pi left(frac{1}{2} r_pright)^2 h )

Simplifying the expression:

( V_{text{glass}} pi left(frac{1}{4} r_p^2right) h frac{1}{4} pi r_p^2 h )

Relating the Volumes

Since we know the volume of the plastic cylinder:

( V_{text{glass}} frac{1}{4} V_{text{plastic}} frac{1}{4} times 64 16 )

Therefore, the volume of the glass cylinder is ( 16 ) cubic units.

Conclusion

The relationship between the volumes of two cylinders with the same height but different radii is straightforward when using the volume formula. As shown, the volume of the glass cylinder is ( frac{1}{4} ) of the volume of the plastic cylinder, given their respective radii.