A box has a base measuring 3 feet by 4 feet. Find the length of the largest rod that can be placed at the bottom of the box.

Answer

To find the length of the largest rod that can be placed at the bottom of a box with a base measuring 3 feet by 4 feet, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Shape**: The base of the box is a rectangle with dimensions 3 feet and 4 feet. We can visualize this rectangle as having corners labeled A, B, C, and D.

2. **Identify the Diagonal**: The largest rod that can fit in the box will be placed along the diagonal of the rectangle. This diagonal will connect two opposite corners of the rectangle (for example, from corner A to corner C).

3. **Use the Pythagorean Theorem**: To calculate the length of the diagonal (which represents the length of the rod), we can use the Pythagorean theorem. According to the theorem, in a right triangle, the square of the length of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the lengths of the other two sides.

\[ \text{Diagonal}^2 = \text{Length}^2 + \text{Width}^2 \]

Here, the length of the rectangle is 4 feet and the width is 3 feet.

4. **Substitute the Values**: Plugging in the values into the equation gives us:

\[ \text{Diagonal}^2 = 4^2 + 3^2 \]

\[ \text{Diagonal}^2 = 16 + 9 \]

\[ \text{Diagonal}^2 = 25 \]

5. **Calculate the Diagonal**: To find the length of the diagonal, we take the square root of both sides:

\[ \text{Diagonal} = \sqrt{25} \]

\[ \text{Diagonal} = 5 \text{ feet} \]

6. **Conclusion**: Therefore, the length of the largest rod that can be placed at the bottom of the box is 5 feet.

### Final Answer:
The length of the largest rod that can be placed at the bottom of the box is **5 feet**.

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