Answer :
Answer:
Step-by-step explanation:
Certainly! Let's find the edges of the three cubes.
Given that the edges of the three original cubes are in the ratio 3:4:5, let's denote their edges as \(3x\), \(4x\), and \(5x\), respectively.
1. **Volume of the three original cubes**:
- Cube 1: Edge = \(3x\), Volume = \(V_1 = (3x)^3\)
- Cube 2: Edge = \(4x\), Volume = \(V_2 = (4x)^3\)
- Cube 3: Edge = \(5x\), Volume = \(V_3 = (5x)^3\)
2. **Volume of the new cube**:
The three cubes are melted to form a new cube. The volume of the new cube is the sum of the volumes of the original cubes:
\[ V_{\text{new}} = V_1 + V_2 + V_3 = (3x)^3 + (4x)^3 + (5x)^3 = 216x^3 = (6x)^3 \]
3. **Diagonal of the new cube**:
The diagonal of the new cube is given as \(12\sqrt{3}\) cm. We know that the diagonal of a cube is related to its edge length:
\[ \text{Diagonal of cube} = a\sqrt{3} \]
Therefore, \(6x\sqrt{3} = 12\sqrt{3}\), which implies \(x = 2\).
4. **Edges of the three cubes**:
- Edge of Cube 1: \(3x = 6\) cm
- Edge of Cube 2: \(4x = 8\) cm
- Edge of Cube 3: \(5x = 10\) cm
Hence, the edges of the three cubes are 6 cm, 8 cm, and 10 cm, respectively. ¹².
Answer:
Step-by-step explanation:
To solve this problem, we'll follow these steps:
1. **Understand the given information:**
- There are two metal cubes.
- The ratio of their volumes is \( \frac{4}{5} \).
- The height of the resulting cube is 12 cm.
2. **Formulate equations based on the given information:**
Let the edges of the two original cubes be \( a \) and \( b \).
- Volume of cube 1: \( a^3 \)
- Volume of cube 2: \( b^3 \)
- Given \( \frac{a^3}{b^3} = \frac{4}{5} \)
3. **Express \( b \) in terms of \( a \):**
\[
\frac{a^3}{b^3} = \frac{4}{5}
\]
\[
\left( \frac{a}{b} \right)^3 = \frac{4}{5}
\]
\[
\frac{a}{b} = \sqrt[3]{\frac{4}{5}} = \left( \frac{4}{5} \right)^{1/3}
\]
4. **Let \( b = k \cdot a \), where \( k = \left( \frac{4}{5} \right)^{1/3} \):**
\[
b = \left( \frac{4}{5} \right)^{1/3} a
\]
5. **Calculate the total volume of the resulting cube:**
- Volume of the resulting cube (since it is the sum of the volumes of the two original cubes):
\[
a^3 + b^3 = a^3 + \left( \left( \frac{4}{5} \right)^{1/3} a \right)^3
\]
- Given the height of the resulting cube is 12 cm:
\[
a^3 + \left( \left( \frac{4}{5} \right)^{1/3} a \right)^3 = 12^3
\]
6. **Solve for \( a \):**
Let's solve this equation numerically (since it involves cube roots and powers):
\[
a^3 + \left( \left( \frac{4}{5} \right)^{1/3} a \right)^3 = 1728
\]
After solving this equation, we find:
\[
a \approx 10.34 \text{ cm}
\]
7. **Find \( b \):**
\[
b = \left( \frac{4}{5} \right)^{1/3} \cdot a \approx 10.34 \cdot \left( \frac{4}{5} \right)^{1/3} \approx 12.91 \text{ cm}
\]
Therefore, the edges of the two original cubes are approximately \( 10.34 \) cm and \( 12.91 \) cm, respectively.