A cubical block of metal weighs 6 pounds. How much will another cube of the same metal weigh if its sides are twice as long?

Options

A. A) 48
B. B) 12
C. C) 36
D. D) 60

Answer

To solve the problem of how much a cubical block of metal will weigh if its sides are twice as long, we can follow these steps:

### Step 1: Understand the relationship between the side length and volume of a cube.
The volume \( V \) of a cube is given by the formula:
\[ V = a^3 \]
where \( a \) is the length of a side of the cube.

### Step 2: Calculate the volume of the original cube.
Let the side length of the original cube be \( a \). The volume of the original cube is:
\[ V_{\text{original}} = a^3 \]

### Step 3: Determine the side length of the new cube.
If the sides of the new cube are twice as long, then the side length of the new cube is:
\[ 2a \]

### Step 4: Calculate the volume of the new cube.
The volume of the new cube is:
\[ V_{\text{new}} = (2a)^3 = 8a^3 \]

### Step 5: Relate the weights of the cubes.
Since the weight of an object is proportional to its volume (assuming the same material and density), we can set up the following relationship:
\[ \frac{\text{Weight of new cube}}{\text{Weight of original cube}} = \frac{V_{\text{new}}}{V_{\text{original}}} \]
Substituting the volumes from steps 2 and 4:
\[ \frac{\text{Weight of new cube}}{6 \text{ pounds}} = \frac{8a^3}{a^3} \]

### Step 6: Simplify the equation.
This simplifies to:
\[ \frac{\text{Weight of new cube}}{6} = 8 \]

### Step 7: Solve for the weight of the new cube.
Multiplying both sides by 6 gives:
\[ \text{Weight of new cube} = 6 \times 8 = 48 \text{ pounds} \]

### Final Answer:
The weight of the new cube is **48 pounds**.

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