this is a optimization problem
A construction company has been offered a contract for $7.8 million to construct
and operate a trucking route for five years to transport ore from a mine site to a
smelter. The smelter is located on a major highway, and the mine is 3 km into a heavily forested area off the road.
Construction (capital) costs are estimated as follows:
•Repaving the highway will cost $200 000km.
•A new gravel road from the mine to the highway will cost $500 000km.
Operating conditions are as follows:
•There will be 100 round trips each day, for 300 days a year, for each of the
five years the mine will be open.
•Operating costs on the gravel road will be $65h, and the speed limit will be
40 kmh.
•Operating costs on the highway will be $50h, and the speed limit will be
70 kmh.
Use calculus to determine if the company should accept the contract. Determine
the average speeds of the trucks along the paved and gravel roads that produce
optimum conditions (maximum profit). What is the maximum profit?
To solve this optimization problem, we need to find the average speeds of the trucks along the paved and gravel roads that produce optimum conditions and determine the maximum profit.
Let's break down the steps to solve this problem:
Step 1: Establish the objective function
The objective is to maximize profit. We need to determine the revenue and cost components of the profit function.
Profit = Revenue - Cost
Step 2: Determine the revenue
The revenue is based on the contract amount of $7.8 million. Since there will be 100 round trips each day for 300 days a year, the total revenue can be calculated as follows:
Revenue = (Number of round trips per day) * (Number of days per year) * (Contract amount)
Revenue = (100 round trips/day) * (300 days/year) * ($7.8 million)
Step 3: Calculate the costs
The costs consist of construction costs and operating costs.
Construction costs:
The cost of repaving the highway is $200,000 per km.
The cost of building a new gravel road is $500,000 per km.
Operating costs:
On the gravel road, the cost is $65 per hour, and the speed limit is 40 km/h.
On the highway, the cost is $50 per hour, and the speed limit is 70 km/h.
Step 4: Determine the average speeds for maximum profit
To find the average speeds that will result in maximum profit, we need to use calculus. We need to find the maximum of the profit function by taking the derivative with respect to the average speed and setting it equal to zero.
Step 5: Calculate the profit
Plug the average speeds found in Step 4 into the profit function to calculate the maximum profit.
By following these steps, you will be able to solve the optimization problem and determine if the construction company should accept the contract, the average speeds of the trucks along the paved and gravel roads that produce optimum conditions, and the maximum profit.