A solid cube with an edge 10 cm is melted to from two equal cubes. Find the ratio of the edge of the smaller cube to the edge of the bigger cube ?
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To find the ratio of the edge of the smaller cube to the edge of the bigger cube, we need to determine the volume of each cube.
1. Start with the volume of the larger cube:
- The volume of a cube is given by the formula V = edge^3.
- In this case, the edge of the larger cube is 10 cm.
- Therefore, the volume of the larger cube is V = 10^3 = 1000 cm^3.
2. Next, find the volume of one of the smaller cubes:
- Let's assume the edge length of the smaller cube is 'x' cm.
- Since we are splitting the larger cube into two equal smaller cubes, both smaller cubes will have the same edge length.
- Therefore, the volume of each smaller cube is V = x^3.
3. Since we have two equal smaller cubes, the total volume of the smaller cubes combined is twice the volume of one of the smaller cubes:
- Total volume of the smaller cubes = 2 * V = 2 * x^3.
4. We know that the volume of the larger cube is 1000 cm^3, and the total volume of the smaller cubes is 2 * x^3. We can set up an equation:
- 2 * x^3 = 1000 cm^3
5. Solve the equation to find the value of 'x':
- Divide both sides of the equation by 2: x^3 = 500 cm^3
- Take the cube root of both sides: x = ∛(500) ≈ 7.937 cm (rounded to three decimal places).
6. Finally, calculate the ratio of the edge length of the smaller cube to the edge length of the larger cube:
- Ratio = (edge length of smaller cube) / (edge length of larger cube) = x / 10
- Substituting the value of 'x' we found: Ratio ≈ 7.937 / 10 ≈ 0.794
Therefore, the ratio of the edge length of the smaller cube to the edge length of the larger cube is approximately 0.794.