To determine the average cost of a gallon of fuel across the trucking company's operations, we need to calculate the expected value of the fuel cost based on the given probabilities and fuel costs for each region. Here’s the step-by-step solution:
### Step-by-Step Calculation:
1. Identify the probability and cost for each region:
- Southeast: Probability [tex]\( P_{SE} = 0.20 \)[/tex], Cost [tex]\( C_{SE} = \$3.10 \)[/tex]
- Southwest: Probability [tex]\( P_{SW} = 0.30 \)[/tex], Cost [tex]\( C_{SW} = \$3.50 \)[/tex]
- California: Probability [tex]\( P_{CA} = 0.50 \)[/tex], Cost [tex]\( C_{CA} = \$4.05 \)[/tex]
2. Calculate the contribution to the expected cost from each region:
- Southeast: [tex]\( 0.20 \times \$3.10 = \$0.62 \)[/tex]
- Southwest: [tex]\( 0.30 \times \$3.50 = \$1.05 \)[/tex]
- California: [tex]\( 0.50 \times \$4.05 = \$2.025 \)[/tex]
3. Sum these contributions to find the total expected cost:
[tex]\[
\text{Expected Cost} = 0.62 + 1.05 + 2.025 = 3.695
\][/tex]
4. Round the result to the nearest cent:
[tex]\[
\text{Rounded Expected Cost} = \$3.70
\][/tex]
### Summary:
The company should budget, on average, \[tex]$3.70 per gallon of fuel across its operations.
### Final Answer:
\[
\boxed{\$[/tex]3.70}
\]
