Let's solve the problem step-by-step:
1. Given Values:
- [tex]\( b = 60 \)[/tex]
- [tex]\( \Delta b = 6 \)[/tex]
- [tex]\( c = 80 \)[/tex]
- [tex]\( \Delta c = 4 \)[/tex]
2. Value of F:
- [tex]\( F = b \cdot c \)[/tex]
- Substituting the values of [tex]\( b \)[/tex] and [tex]\( c \)[/tex]:
[tex]\[
F = 60 \cdot 80 = 4800
\][/tex]
3. Absolute Errors:
- The absolute error in [tex]\( b \)[/tex] is [tex]\( \Delta b = 6 \)[/tex]
- The absolute error in [tex]\( c \)[/tex] is [tex]\( \Delta c = 4 \)[/tex]
4. Relative Errors:
- The relative error in [tex]\( b \)[/tex]:
[tex]\[
\frac{\Delta b}{b} = \frac{6}{60} = 0.1
\][/tex]
- The relative error in [tex]\( c \)[/tex]:
[tex]\[
\frac{\Delta c}{c} = \frac{4}{80} = 0.05
\][/tex]
5. Relative Error in F:
- The relative error in [tex]\( F \)[/tex] is the sum of the relative errors of [tex]\( b \)[/tex] and [tex]\( c \)[/tex]:
[tex]\[
\frac{\Delta F}{F} = 0.1 + 0.05 = 0.15
\][/tex]
6. Absolute Error in F:
- The absolute error in [tex]\( F \)[/tex] is found by multiplying the relative error by the value of [tex]\( F \)[/tex]:
[tex]\[
\Delta F = \left( \frac{\Delta F}{F} \right) \cdot F = 0.15 \cdot 4800 = 720
\][/tex]
7. Percentage Error in F:
- To find the percentage error in [tex]\( F \)[/tex], convert the relative error to a percentage by multiplying by 100:
[tex]\[
\% \Delta F = 0.15 \cdot 100 = 15\%
\][/tex]
So, the results are:
- The value of [tex]\( F \)[/tex] is 4800.
- The absolute error in [tex]\( F \)[/tex] is 720.
- The relative error in [tex]\( F \)[/tex] is 0.15.
- The percentage error in [tex]\( F \)[/tex] is 15\%.
