Merging Cubes: Theory and Calculation
When working with three dimensional objects, especially cubes, certain principles of geometry play a key role. This article delves into the concept of melting and recasting cubes into a new cube, a problem often encountered in middle and high school mathematics. We will demonstrate how the volumes and surface areas of the original and new cubes are related and provide a step-by-step guide to solving such problems effectively.
The Problem at Hand
Consider three cubes with edge lengths of 3 cm, 4 cm, and 5 cm. If these cubes are melted and recast into a single new cube, what is the surface area of the new cube? This problem is typical in geometry and can be solved by understanding the principles of volume conservation and surface area calculation.
Step-by-Step Solution
The first step is to calculate the volumes of the three original cubes. The volume of a cube is calculated using the formula V side^3, where the side is the length of one edge of the cube.
The volume of the first cube (edge 3 cm):
V1 3 cm * 3 cm * 3 cm 27 cm3
The volume of the second cube (edge 4 cm):
V2 4 cm * 4 cm * 4 cm 64 cm3
The volume of the third cube (edge 5 cm):
V3 5 cm * 5 cm * 5 cm 125 cm3
Since the volume of a substance remains constant during melting and recasting, the volume of the new cube is the sum of the volumes of the three original cubes.
Therefore, the total volume (V) of the new cube is:
V 27 cm3 64 cm3 125 cm3 216 cm3
To find the edge length of the new cube, we take the cube root of its volume:
a (216 cm3)1/3 6 cm
Now that we have the edge length of the new cube, we can calculate its surface area using the formula for the surface area of a cube: TSA 6 * (side^2).
Hence, the surface area of the new cube is:
TSA 6 * (6 cm)^2 6 * 36 cm2 216 cm2
Additional Examples and Calculations
Consider another set of cubes with different edge lengths. For example, cubes with edges of 3 cm, 6 cm, and 8 cm have volumes as follows:
The volume of the first cube (edge 3 cm):
V1 3 cm * 3 cm * 3 cm 27 cm3
The volume of the second cube (edge 6 cm):
V2 6 cm * 6 cm * 6 cm 216 cm3
The volume of the third cube (edge 8 cm):
V3 8 cm * 8 cm * 8 cm 512 cm3
The total volume of the new cube is the sum of these volumes:
V 27 cm3 216 cm3 512 cm3 755 cm3
The edge length (a) of the new cube is:
a (755 cm3)1/3 ≈ 9.093 cm
The surface area (TSA) of the new cube is:
TSA 6 * (9.093 cm)^2 ≈ 497.49 cm2
Conclusion
Understanding the principles of volume and surface area calculations is crucial in solving problems related to the merging and recasting of geometric shapes, such as cubes. By applying these principles, one can easily determine the surface area of the new cube formed by recasting the original cubes.