Answer :
To determine the required precision for thermometers that measure both Fahrenheit and Celsius, we need to match the given precision values correctly. Here’s how you can solve this question step-by-step:
1. Understand the Choices:
The question provides four pairs of precision values:
- [tex]\( \pm 0.9^{\circ} F \left( \pm 0.5^{\circ} C \right) \)[/tex]
- [tex]\( \pm 1.8^{\circ} F \left( \pm 1^{\circ} C \right) \)[/tex]
- [tex]\( \pm 2.7^{\circ} F \left( \pm 1.5^{\circ} C \right) \)[/tex]
- [tex]\( \pm 3.6^{\circ} F \left( \pm 2^{\circ} C \right) \)[/tex]
2. Celsius and Fahrenheit Relationship:
The relationship between Fahrenheit ([tex]\(F\)[/tex]) and Celsius ([tex]\(C\)[/tex]) is given by the formula:
[tex]\[
F = C \times 1.8 + 32
\][/tex]
However, for small changes in temperature, the conversion factor is [tex]\(1.8\)[/tex]. Therefore, an error of [tex]\( \pm 1^{\circ} C \)[/tex] translates to [tex]\(1.8 \times 1^{\circ} C = \pm 1.8^{\circ} F\)[/tex].
3. Match Precisions:
Compare each given precision pair to find which pairs correctly correspond to the conversion factor:
- [tex]\( \pm 0.9^{\circ} F \)[/tex] should correspond to [tex]\( \pm 0.5^{\circ} C \)[/tex].
- For verification: [tex]\(0.5^{\circ} C \times 1.8 = 0.9^{\circ} F\)[/tex] (Correct).
- [tex]\( \pm 1.8^{\circ} F \)[/tex] should correspond to [tex]\( \pm 1^{\circ} C \)[/tex].
- For verification: [tex]\(1^{\circ} C \times 1.8 = 1.8^{\circ} F\)[/tex] (Correct).
- [tex]\( \pm 2.7^{\circ} F \)[/tex] should correspond to [tex]\( \pm 1.5^{\circ} C \)[/tex].
- For verification: [tex]\(1.5^{\circ} C \times 1.8 = 2.7^{\circ} F\)[/tex] (Correct).
- [tex]\( \pm 3.6^{\circ} F \)[/tex] should correspond to [tex]\( \pm 2^{\circ} C \)[/tex].
- For verification: [tex]\(2^{\circ} C \times 1.8 = 3.6^{\circ} F\)[/tex] (Correct).
4. Determine the Correct Answer:
All the choices align correctly if we match them with the conversion factor. So, to select one, we need to choose the pair with the smallest acceptable tolerance, which minimizes the possible error in temperature readings.
Based on the verification process:
- The pair with the smallest tolerances of [tex]\(\pm 0.9^{\circ} F \left( \pm 0.5^{\circ} C \right)\)[/tex] offers the highest precision.
Thus, the correct answer is:
[tex]\[
1. \ \pm 0.9^{\circ} F \left( \pm 0.5^{\circ} C \right)
\][/tex]
1. Understand the Choices:
The question provides four pairs of precision values:
- [tex]\( \pm 0.9^{\circ} F \left( \pm 0.5^{\circ} C \right) \)[/tex]
- [tex]\( \pm 1.8^{\circ} F \left( \pm 1^{\circ} C \right) \)[/tex]
- [tex]\( \pm 2.7^{\circ} F \left( \pm 1.5^{\circ} C \right) \)[/tex]
- [tex]\( \pm 3.6^{\circ} F \left( \pm 2^{\circ} C \right) \)[/tex]
2. Celsius and Fahrenheit Relationship:
The relationship between Fahrenheit ([tex]\(F\)[/tex]) and Celsius ([tex]\(C\)[/tex]) is given by the formula:
[tex]\[
F = C \times 1.8 + 32
\][/tex]
However, for small changes in temperature, the conversion factor is [tex]\(1.8\)[/tex]. Therefore, an error of [tex]\( \pm 1^{\circ} C \)[/tex] translates to [tex]\(1.8 \times 1^{\circ} C = \pm 1.8^{\circ} F\)[/tex].
3. Match Precisions:
Compare each given precision pair to find which pairs correctly correspond to the conversion factor:
- [tex]\( \pm 0.9^{\circ} F \)[/tex] should correspond to [tex]\( \pm 0.5^{\circ} C \)[/tex].
- For verification: [tex]\(0.5^{\circ} C \times 1.8 = 0.9^{\circ} F\)[/tex] (Correct).
- [tex]\( \pm 1.8^{\circ} F \)[/tex] should correspond to [tex]\( \pm 1^{\circ} C \)[/tex].
- For verification: [tex]\(1^{\circ} C \times 1.8 = 1.8^{\circ} F\)[/tex] (Correct).
- [tex]\( \pm 2.7^{\circ} F \)[/tex] should correspond to [tex]\( \pm 1.5^{\circ} C \)[/tex].
- For verification: [tex]\(1.5^{\circ} C \times 1.8 = 2.7^{\circ} F\)[/tex] (Correct).
- [tex]\( \pm 3.6^{\circ} F \)[/tex] should correspond to [tex]\( \pm 2^{\circ} C \)[/tex].
- For verification: [tex]\(2^{\circ} C \times 1.8 = 3.6^{\circ} F\)[/tex] (Correct).
4. Determine the Correct Answer:
All the choices align correctly if we match them with the conversion factor. So, to select one, we need to choose the pair with the smallest acceptable tolerance, which minimizes the possible error in temperature readings.
Based on the verification process:
- The pair with the smallest tolerances of [tex]\(\pm 0.9^{\circ} F \left( \pm 0.5^{\circ} C \right)\)[/tex] offers the highest precision.
Thus, the correct answer is:
[tex]\[
1. \ \pm 0.9^{\circ} F \left( \pm 0.5^{\circ} C \right)
\][/tex]
