16. Given the following information, determine the constant speed, travel time, and cost
that represent the minimum one-way trip delivery cost.
Fuel costs per hour are equal to V^2/600
V is the speed in miles per hour
Driver (labor) costs are $10 per hour
One-way distance is 110 miles
To determine the constant speed, travel time, and cost that represent the minimum one-way trip delivery cost, we need to consider both the fuel costs and the driver (labor) costs.
First, let's calculate the fuel costs:
The fuel costs per hour are equal to V^2/600, where V is the speed in miles per hour.
Since we want to minimize the delivery cost, we need to minimize the fuel costs.
Next, let's calculate the travel time:
We know that the one-way distance is 110 miles.
The travel time is equal to the distance divided by the speed:
Travel time = Distance / Speed = 110 miles / V miles per hour
Now, let's calculate the driver costs:
The driver costs are a fixed rate of $10 per hour.
To determine the minimum delivery cost, we need to consider the minimum fuel costs and the cost of the driver for the travel time.
To find the minimum fuel costs, we need to find the minimum value of V^2/600. Since V^2/600 is a quadratic function, we can find the minimum by taking the derivative and setting it equal to zero.
d/dV (V^2/600) = 0
2V/600 = 0
V = 0
Since we are dealing with speeds, a speed of 0 is not possible, so we can ignore this result.
Now, let's calculate the minimum travel time and cost by plugging the minimum fuel costs into the formulas we derived earlier.
Travel time = 110 miles / V miles per hour
Driver costs = $10 per hour
Fuel costs = V^2/600
Let's calculate the values for a few speeds:
For V = 50 miles per hour:
Travel time = 110 miles / 50 miles per hour = 2.2 hours
Driver costs = $10 per hour
Fuel costs = (50^2)/600 = 4.17
For V = 60 miles per hour:
Travel time = 110 miles / 60 miles per hour ≈ 1.83 hours
Driver costs = $10 per hour
Fuel costs = (60^2)/600 = 6
For V = 70 miles per hour:
Travel time = 110 miles / 70 miles per hour ≈ 1.57 hours
Driver costs = $10 per hour
Fuel costs = (70^2)/600 ≈ 8.17
By comparing the costs for different speeds, you can determine which speed, travel time, and cost represent the minimum one-way trip delivery cost. In this case, the calculations show that the minimum cost occurs at a speed of 70 miles per hour. Therefore, the constant speed is 70 miles per hour, the travel time is approximately 1.57 hours, and the total cost is the sum of the driver costs and fuel costs, which would be $10 + 8.17 = $18.17.