A cube of metal , each each of which measures 4 cm, weighs 400 kgs. What is the length of each edge of a cube of the same metal which weighs 3200 kg?

Options

A. 64 cm
B. 8 cm
C. 2 cm
D. none of these

Answer

To solve the problem step by step, we will use the relationship between the weight of the cube, its volume, and the side length.

### Step 1: Calculate the volume of the first cube
The volume \( V \) of a cube is given by the formula:
\[ V = \text{side}^3 \]

Given that the side of the first cube is 4 cm:
\[ V_1 = 4^3 = 64 \, \text{cm}^3 \]

### Step 2: Determine the density of the metal
The weight of the first cube is given as 400 kg. We can find the density \( \rho \) of the metal using the formula:
\[ \rho = \frac{\text{weight}}{\text{volume}} \]

Substituting the values we have:
\[ \rho = \frac{400 \, \text{kg}}{64 \, \text{cm}^3} = 6.25 \, \text{kg/cm}^3 \]

### Step 3: Calculate the volume of the second cube
We need to find the volume of the cube that weighs 3200 kg. Using the density we calculated:
\[ \text{Volume} = \frac{\text{weight}}{\text{density}} \]
\[ V_2 = \frac{3200 \, \text{kg}}{6.25 \, \text{kg/cm}^3} \]

Calculating this gives:
\[ V_2 = 512 \, \text{cm}^3 \]

### Step 4: Find the side length of the second cube
Now, we can find the side length of the second cube using the volume formula:
\[ V = \text{side}^3 \]
So, we need to solve for the side:
\[ \text{side} = V^{1/3} \]
\[ \text{side} = (512)^{1/3} \]

Calculating this gives:
\[ \text{side} = 8 \, \text{cm} \]

### Conclusion
The length of each edge of the cube that weighs 3200 kg is **8 cm**.

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