Two rectangular boxes are similar.One box is larger by a scale factor of 2.The smaller box has dimensions 1 metre, 2 metres and 3 metres.

1. Break Down the Problem

We have two similar rectangular boxes. The smaller box has dimensions:

• Length: $1$ metre
• Width: $2$ metres
• Height: $3$ metres

The larger box is scaled by a factor of $2$. We need to find the dimensions of the larger box.

2. Relevant Concepts

For similar geometric shapes, corresponding dimensions scale by the same factor. Therefore, if the smaller box has dimensions $L$, $W$, and $H$, the larger box dimensions can be represented as:

• $L' = k \cdot L$
• $W' = k \cdot W$
• $H' = k \cdot H$

Where $k$ is the scale factor (in this case, $k = 2$).

3. Analysis and Detail

Let's calculate the dimensions of the larger box.

• Length of the larger box: $L' = 2 \cdot L = 2 \cdot 1 = 2 \text{ metres}$

• Width of the larger box: $W' = 2 \cdot W = 2 \cdot 2 = 4 \text{ metres}$

• Height of the larger box: $H' = 2 \cdot H = 2 \cdot 3 = 6 \text{ metres}$

4. Verify and Summarize

We have calculated all the dimensions of the larger box based on the scaling factor of $2$. The calculations have been double-checked for correctness.

Final Answer

The dimensions of the larger box are:

• Length: $2$ metres
• Width: $4$ metres
• Height: $6$ metres