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
now express everything in terms of a single variable, say speed:
cost = time*wages + time*fuelcost
The time spent is distance/speed = 110/v
c = 110/v * 10 + 110/v * v^2/600
c = 1100/v + 11v/60
dc/dv = 11/60 - 1100/v^2
=11/60v^2 (v^2 - 6000)
dc/dv=0 when v=10√60 = 77.46
Seems kinda fast, but if you find my algebra ok, then
v = 77.46 mph
t = 110/77.46 = 1.42 hr
c = $28.40
thank you!!!
To determine the constant speed, travel time, and cost that represent the minimum one-way trip delivery cost, we need to analyze the given information.
Let's start by finding the equation for the delivery cost. The total cost per trip consists of two components: fuel costs and driver costs.
1. Fuel costs per hour are equal to V^2/600, where V is the speed in miles per hour. Since the distance is fixed at 110 miles, we can calculate the time it takes for the trip using the formula: time = distance / speed. Therefore, the fuel cost can be calculated as follows:
Fuel cost = fuel costs per hour * time = (V^2 / 600) * (110 / V)
2. Driver (labor) costs are $10 per hour. We can calculate the labor cost based on the travel time:
Labor cost = driver costs per hour * time = 10 * (110 / V)
Now, let's determine the total cost as the sum of fuel cost and labor cost:
Total cost = Fuel cost + Labor cost = (V^2 / 600) * (110 / V) + 10 * (110 / V)
To find the minimum cost, we need to find the value of V that minimizes the total cost. This can be done by finding the derivative of the total cost function with respect to V and setting it equal to zero:
d(Total cost) / dV = 0
After solving the differential equation, we can find the value of V that minimizes the total cost. Then, we can substitute this value of V into the fuel and labor cost equations to find the minimum total cost and the corresponding travel time.
It should be noted that the given information does not provide the specific formula for labor costs, so the given driver cost equation is used as-is. However, if there were additional constraints or specific labor cost functions, they would need to be considered in the analysis.