To find the upper bound of the value of [tex]\(X\)[/tex], given the formula [tex]\( X = \frac{2a - b}{f} \)[/tex], where:
- [tex]\( a = 7.5 \)[/tex] correct to 1 decimal place,
- [tex]\( b = 3.42 \)[/tex] correct to 2 decimal places,
- [tex]\( f = 2 \)[/tex] correct to the nearest whole number,
we need to determine the upper and lower bounds of each variable.
### Step 1: Determine the bounds for [tex]\( a \)[/tex]
[tex]\[
a = 7.5 \quad \text{(1 decimal place)}
\][/tex]
- Lower bound of [tex]\( a \)[/tex]: [tex]\( 7.45 \)[/tex]
- Upper bound of [tex]\( a \)[/tex]: [tex]\( 7.55 \)[/tex]
### Step 2: Determine the bounds for [tex]\( b \)[/tex]
[tex]\[
b = 3.42 \quad \text{(2 decimal places)}
\][/tex]
- Lower bound of [tex]\( b \)[/tex]: [tex]\( 3.415 \)[/tex]
- Upper bound of [tex]\( b \)[/tex]: [tex]\( 3.425 \)[/tex]
### Step 3: Determine the bounds for [tex]\( f \)[/tex]
[tex]\[
f = 2 \quad \text{(nearest whole number)}
\][/tex]
- Lower bound of [tex]\( f \)[/tex]: [tex]\( 1.5 \)[/tex]
- Upper bound of [tex]\( f \)[/tex]: [tex]\( 2.5 \)[/tex]
### Step 4: Calculate intermediate values using the bounds
To find the upper bound of [tex]\( X \)[/tex], we use:
- Upper bound for [tex]\( a \)[/tex]
- Lower bound for [tex]\( b \)[/tex]
- Lower bound for [tex]\( f \)[/tex]
These choices maximize the numerator and minimize the denominator.
#### Upper bound for [tex]\( a \)[/tex]:
[tex]\[
a_{\text{upper}} = 7.55
\][/tex]
#### Lower bound for [tex]\( b \)[/tex]:
[tex]\[
b_{\text{lower}} = 3.415
\][/tex]
#### Lower bound for [tex]\( f \)[/tex]:
[tex]\[
f_{\text{lower}} = 1.5
\][/tex]
### Step 5: Substitute these bounds into the formula to find the upper bound of [tex]\( X \)[/tex]
[tex]\[
X_{\text{upper}} = \frac{2 \cdot a_{\text{upper}} - b_{\text{lower}}}{f_{\text{lower}}}
\][/tex]
First, calculate [tex]\( 2 \cdot a_{\text{upper}} \)[/tex]:
[tex]\[
2 \cdot 7.55 = 15.1
\][/tex]
Next, calculate the numerator [tex]\( 2 \cdot a_{\text{upper}} - b_{\text{lower}} \)[/tex]:
[tex]\[
15.1 - 3.415 = 11.685
\][/tex]
Finally, divide by the lower bound of [tex]\( f \)[/tex]:
[tex]\[
X_{\text{upper}} = \frac{11.685}{1.5} = 7.79
\][/tex]
### Final Answer:
The upper bound for the value of [tex]\( X \)[/tex] is approximately [tex]\( 7.79 \)[/tex].
