The ratio of the length to the breadth to the height of a box is 7:5:4. If the total length of all the edges of the box is 448cm, what is the volume of the box?

Given that the ratio of the length, breadth, and height of the box is \(7:5:4\), and the total length of all the edges of the box is 448 cm, we are asked to find the volume of the box.

### Step 1: Express dimensions in terms of a common factor
Let:
- \( l = 7x \) be the length,
- \( b = 5x \) be the breadth,
- \( h = 4x \) be the height.

Here, \(x\) is a common factor to be determined.

### Step 2: Use the formula for the total length of all edges of a box
The formula for the total length of the edges of a rectangular box is:
\[
4(l + b + h)
\]
Substituting the expressions for \(l\), \(b\), and \(h\):
\[
4(7x + 5x + 4x) = 448
\]
Simplifying:
\[
4(16x) = 448
\]
\[
64x = 448
\]
Solving for \(x\):
\[
x = \frac{448}{64} = 7
\]

### Step 3: Calculate the actual dimensions
Now that we have \(x = 7\), we can calculate the actual dimensions:
- Length: \( l = 7x = 7 \times 7 = 49 \, \text{cm} \)
- Breadth: \( b = 5x = 5 \times 7 = 35 \, \text{cm} \)
- Height: \( h = 4x = 4 \times 7 = 28 \, \text{cm} \)

### Step 4: Calculate the volume
The volume \(V\) of a rectangular box is given by the formula:
\[
V = l \times b \times h
\]
Substituting the values:
\[
V = 49 \times 35 \times 28
\]
Now calculating the volume:
\[
V = 49 \times 35 = 1715
\]
\[
V = 1715 \times 28 = 48020 \, \text{cm}^3
\]

Thus, the volume of the box is \(48,020 \, \text{cm}^3\).

Would you like further details or explanations?

Here are 5 related questions to further your understanding:
1. How is the formula for the total length of all edges of a box derived?
2. How would the volume change if the ratio of the dimensions was altered but the total length of the edges remained the same?
3. Can you derive a formula to directly compute the volume from the edge length and dimension ratios?
4. What happens to the volume if each dimension of the box is doubled?
5. How is the concept of ratio applied in three-dimensional geometry?

**Tip**: Always double-check your calculations, especially when working with large numbers or multiple steps in geometry!

Learn how to calculate the volume of a box where the ratio of its dimensions is 7:5:4, and the total length of all edges is 448 cm. This step-by-step guide covers key concepts in geometry, ratios, and algebra. Suitable for students in Grades 7-9.

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