Sure! To calculate these expressions, we need to consider both the values and uncertainties of the given quantities. The uncertainties are represented by the ± symbol.
Here's how to calculate each expression step by step:
1. a + b:
- Add the values of a and b: 40 + 30 = 70
- Add the uncertainties: 5 + 3 = 8
- The result is (70 ± 8)
2. a + b + c:
- Add the values of a, b, and c: 40 + 30 + 20 = 90
- Add the uncertainties: 5 + 3 + 1 = 9
- The result is (90 ± 9)
3. a / t:
- Divide the value of a by the value of t: 40 / 1.2 = 33.33...
- Calculate the fractional uncertainty: (uncertainty of a / value of a) + (uncertainty of t / value of t) = (5/40) + (0.1/1.2) = 0.125 + 0.083... = 0.208...
- Multiply the value by the fractional uncertainty: 33.33... * 0.208... = 6.94...
- The result is (33.33... ± 6.94...)
4. (a + c) / t:
- Add the values of a and c: 40 + 20 = 60
- Divide the result by the value of t: 60 / 1.2 = 50
- Calculate the fractional uncertainty: (uncertainty of a / value of a) + (uncertainty of c / value of c) + (uncertainty of t / value of t) = (5/40) + (1/20) + (0.1/1.2) = 0.125 + 0.05 + 0.083... = 0.258...
- Multiply the value by the fractional uncertainty: 50 * 0.258... = 12.92...
- The result is (50 ± 12.92...)
5. (1.23 ± 0.03) + pi:
- Add the value of 1.23 and the value of pi: 1.23 + π
- Add the uncertainties: 0.03
- The result is (1.23 + π ± 0.03)
6. (1.23 ± 0.03) × pi:
- Multiply the value of 1.23 by pi: 1.23 × π
- Calculate the fractional uncertainty: (uncertainty / value) = 0.03 / 1.23 ≈ 0.024...
- Multiply the value by the fractional uncertainty: (1.23 × π) * 0.024... ≈ 0.076...
- The result is (1.23 × π ± 0.076...)
I hope this helps! Let me know if you have any further questions.