the weights of 5 boxes of srews vary from 2.85 pounds to 3.45 pounds. If w represents the weight, in pounds, of one of these boxes, which of the following must be true?

Options

A. `|w-2.85|le0.3`
B. `|w-3.15|le0.3`
C. `|w-5|le0.3`
D. `|w-0.3|le3.15`

Answer

To solve the problem, we need to determine the range of weights for the boxes of screws given that the weights vary from 2.85 pounds to 3.45 pounds. We will represent this mathematically and analyze the inequalities involved.

### Step-by-Step Solution:

1. **Identify the Range of Weights**:
The weights of the boxes are given to vary from 2.85 pounds to 3.45 pounds. This can be expressed as:
\[ 2.85 \leq w \leq 3.45 \]

2. **Rewrite the Inequality**:
We can rewrite the inequality in terms of a central value and a deviation. The midpoint (average) of the range can be calculated as:
\[ \text{Midpoint} = \frac{2.85 + 3.45}{2} = \frac{6.30}{2} = 3.15 \]

3. **Calculate the Deviation**:
Next, we calculate the deviation from the midpoint to the endpoints of the range:
\[ \text{Deviation} = 3.45 - 3.15 = 0.30 \]
\[ \text{Deviation} = 3.15 - 2.85 = 0.30 \]
Thus, the maximum deviation from the midpoint is 0.30 pounds.

4. **Set Up the Absolute Value Inequality**:
We can express the range of weights using the absolute value notation:
\[ |w - 3.15| \leq 0.30 \]
This inequality states that the weight \( w \) can deviate from 3.15 pounds by at most 0.30 pounds.

5. **Conclusion**:
Therefore, the inequality \( |w - 3.15| \leq 0.30 \) must be true for the weights of the boxes of screws.

### Final Answer:
The statement that must be true is:
\[ |w - 3.15| \leq 0.30 \]