To determine the range of acceptable lengths for the metal rod, we start by identifying the ideal length and the permissible tolerance. The ideal length of the rod is given as [tex]\(35.5 \, \text{cm}\)[/tex], and the length can vary by at most [tex]\(0.035 \, \text{cm}\)[/tex] from this ideal length.
Step-by-Step Solution:
1. Identify the Ideal Length and Tolerance:
- Ideal length ([tex]\(L\)[/tex]) = [tex]\(35.5 \, \text{cm}\)[/tex]
- Tolerance ([tex]\(T\)[/tex]) = [tex]\(0.035 \, \text{cm}\)[/tex]
2. Calculate the Lower Bound of Acceptable Lengths:
- The lower bound can be found by subtracting the tolerance from the ideal length:
[tex]\[
\text{Lower Bound} = L - T = 35.5 \, \text{cm} - 0.035 \, \text{cm} = 35.465 \, \text{cm}
\][/tex]
3. Calculate the Upper Bound of Acceptable Lengths:
- The upper bound can be found by adding the tolerance to the ideal length:
[tex]\[
\text{Upper Bound} = L + T = 35.5 \, \text{cm} + 0.035 \, \text{cm} = 35.535 \, \text{cm}
\][/tex]
4. Write the Acceptable Range:
- The rod's length must lie within the range inclusive of these bounds:
[tex]\[
35.465 \leq x \leq 35.535
\][/tex]
Thus, the correct range of acceptable lengths for the rod is [tex]\(35.465 \leq x \leq 35.535\)[/tex].
Answer:
The correct option is:
D. [tex]\(35.465 \leq x \leq 35.535\)[/tex]
