A shipping company wants to optimize its delivery routes to reduce fuel consumption and overall transportation costs. They currently have 100 trucks, with an average daily route of 200 miles per truck. Each truck consumes 6 miles per gallon of diesel fuel, which costs $3 per gallon. The company is considering investing in new routing software that claims to reduce total mileage by 15%. Calculate the potential annual savings from reduced fuel consumption and determine how long it would take for the investment in the software to pay off if it costs $500,000.

First, let's calculate the total daily mileage for all trucks:

100 trucks * 200 miles/truck = 20,000 miles

Now let's calculate the daily fuel consumption:

20,000 miles / 6 miles/gallon = 3,333.33 gallons

Next, we'll find out how much they spend on fuel per day:

3,333.33 gallons * $3/gallon = $10,000

Now let's calculate their annual fuel cost:

$10,000/day * 365 days/year = $3,650,000/year

If the new routing software reduces total mileage by 15%, then it will reduce fuel consumption and costs by the same percentage. Let's calculate the potential annual savings:

$3,650,000/year * 0.15 = $547,500/year

Now let's determine how long it would take for the investment in the software to pay off if it costs $500,000:

$500,000 / $547,500/year = 0.913 years or approximately 11 months