To determine the average amount the trucking company should budget for a gallon of fuel across its operations, we'll calculate the weighted average of fuel costs based on the given probabilities for each region.
### Step-by-Step Solution:
1. Identify the given data:
- Probability in Southeast ([tex]\(P_{SE}\)[/tex]): 20% or 0.20
- Probability in Southwest ([tex]\(P_{SW}\)[/tex]): 30% or 0.30
- Probability in California ([tex]\(P_{CA}\)[/tex]): 50% or 0.50
- Fuel cost in Southeast ([tex]\(C_{SE}\)[/tex]): [tex]$3.10 per gallon
- Fuel cost in Southwest (\(C_{SW}\)): $[/tex]3.50 per gallon
- Fuel cost in California ([tex]\(C_{CA}\)[/tex]): [tex]$4.05 per gallon
2. Calculate the contribution of each region to the average fuel cost:
- Contribution from Southeast: \(P_{SE} \times C_{SE} = 0.20 \times 3.10 = 0.62\)
- Contribution from Southwest: \(P_{SW} \times C_{SW} = 0.30 \times 3.50 = 1.05\)
- Contribution from California: \(P_{CA} \times C_{CA} = 0.50 \times 4.05 = 2.025\)
3. Sum the contributions to find the weighted average fuel cost:
- Weighted average fuel cost: \(0.62 + 1.05 + 2.025 = 3.695\)
4. Round the weighted average fuel cost to the nearest cent:
- $[/tex]3.695[tex]$ rounded to the nearest cent is $[/tex]3.70
Thus, the amount the company should budget on average for a gallon of fuel across its operations is $3.70.
