Answer :
To determine the lower and upper bounds for the expression \(\frac{a}{b - c}\), considering the given values with their precision constraints, we proceed as follows:
1. Interpreting the precision of given values:
- \( a = 360 \) to 2 significant figures implies the actual value of \( a \) ranges from 355 to 365.
- \( b = 22 \) rounded to the nearest integer implies the actual value of \( b \) ranges from 21.5 to 22.5.
- \( c = 4.4 \) to 1 decimal place implies the actual value of \( c \) ranges from 4.35 to 4.45.
2. Determine the lower bound:
To find the lower bound of \(\frac{a}{b - c}\):
- We use the maximum possible value of \( a \), which is 365.
- We look for the minimum value of the denominator \( b - c \), which occurs when \( b \) is at its minimum and \( c \) is at its maximum.
- Minimum \( b \) is 21.5.
- Maximum \( c \) is 4.45.
Thus, the denominator \( b_{\min} - c_{\max} = 21.5 - 4.45 = 17.05 \).
Therefore, the lower bound calculation is:
[tex]\[ \text{Lower Bound} = \frac{365}{17.05} \][/tex]
3. Determine the upper bound:
To find the upper bound of \(\frac{a}{b - c}\):
- We use the minimum possible value of \( a \), which is 355.
- We look for the maximum value of the denominator \( b - c \), which occurs when \( b \) is at its maximum and \( c \) is at its minimum.
- Maximum \( b \) is 22.5.
- Minimum \( c \) is 4.35.
Thus, the denominator \( b_{\max} - c_{\min} = 22.5 - 4.35 = 18.15 \).
Therefore, the upper bound calculation is:
[tex]\[ \text{Upper Bound} = \frac{355}{18.15} \][/tex]
4. Final calculations and rounding:
- After performing these computations, the lower bound value rounds to 21.4, and the upper bound value rounds to 19.6, both rounded to 1 decimal place.
So, the lower and upper bounds for \(\frac{a}{b - c}\) are:
[tex]\[ \text{Lower Bound} = 21.4 \][/tex]
[tex]\[ \text{Upper Bound} = 19.6 \][/tex]
Thus, the final results are:
[tex]\[ \boxed{21.4 \text{ (lower bound)}, \ 19.6 \text{ (upper bound)}} \][/tex]
1. Interpreting the precision of given values:
- \( a = 360 \) to 2 significant figures implies the actual value of \( a \) ranges from 355 to 365.
- \( b = 22 \) rounded to the nearest integer implies the actual value of \( b \) ranges from 21.5 to 22.5.
- \( c = 4.4 \) to 1 decimal place implies the actual value of \( c \) ranges from 4.35 to 4.45.
2. Determine the lower bound:
To find the lower bound of \(\frac{a}{b - c}\):
- We use the maximum possible value of \( a \), which is 365.
- We look for the minimum value of the denominator \( b - c \), which occurs when \( b \) is at its minimum and \( c \) is at its maximum.
- Minimum \( b \) is 21.5.
- Maximum \( c \) is 4.45.
Thus, the denominator \( b_{\min} - c_{\max} = 21.5 - 4.45 = 17.05 \).
Therefore, the lower bound calculation is:
[tex]\[ \text{Lower Bound} = \frac{365}{17.05} \][/tex]
3. Determine the upper bound:
To find the upper bound of \(\frac{a}{b - c}\):
- We use the minimum possible value of \( a \), which is 355.
- We look for the maximum value of the denominator \( b - c \), which occurs when \( b \) is at its maximum and \( c \) is at its minimum.
- Maximum \( b \) is 22.5.
- Minimum \( c \) is 4.35.
Thus, the denominator \( b_{\max} - c_{\min} = 22.5 - 4.35 = 18.15 \).
Therefore, the upper bound calculation is:
[tex]\[ \text{Upper Bound} = \frac{355}{18.15} \][/tex]
4. Final calculations and rounding:
- After performing these computations, the lower bound value rounds to 21.4, and the upper bound value rounds to 19.6, both rounded to 1 decimal place.
So, the lower and upper bounds for \(\frac{a}{b - c}\) are:
[tex]\[ \text{Lower Bound} = 21.4 \][/tex]
[tex]\[ \text{Upper Bound} = 19.6 \][/tex]
Thus, the final results are:
[tex]\[ \boxed{21.4 \text{ (lower bound)}, \ 19.6 \text{ (upper bound)}} \][/tex]
