A truck driver, on assignment from the owner of the truck is to drive on a 300 mile stretch of highway at a constant speed of v miles per hour. According to road signs, the minimum speed allowed is 55 miles per hour and the speed limit is 70 miles per hour. The cost of gas on the day of the trip is $2.60 per gallon and the gas in the truck has been measured to consumed at a rate of (1+(1/400)*(v^2) gallons per hour.
If the truck driver earns $20 per hour what speed v should the truck driver be assigned to drive in order to keep the cost for the owner of the company as low as possible? Find the minimum cost and compare it to the cost when the driver drives at 55 and 70 miles per hour.
To find the speed at which the truck driver should be assigned to minimize the cost for the owner of the company, we need to compare the costs at different speeds and determine the minimum cost.
Let's first calculate the cost of gas for the trip at constant speed (v) on the 300-mile stretch of highway.
The gas consumption rate is given by (1 + (1/400)*(v^2)) gallons per hour.
To find the total gas consumption, we divide the distance traveled by the speed to get the time:
Time (in hours) = distance (in miles) / speed (in miles per hour)
So, the total gas consumption (in gallons) is:
Gas consumption = Time * gas consumption rate
Substituting the values:
Gas consumption = (300 / v) * (1 + (1/400)*(v^2))
Now, let's calculate the cost of gas:
Cost of gas = Gas consumption (in gallons) * Cost per gallon
Cost of gas = (300 / v) * (1 + (1/400)*(v^2)) * 2.60
Next, let's consider the total cost for the trip, which includes the cost of gas and the truck driver's earnings:
Total cost = Cost of gas + (Time taken for the trip * Earnings per hour)
Time taken for the trip = distance (300 miles) / speed (v miles per hour)
Total cost = Cost of gas + ((300 / v) * 20)
Now, we need to find the value of v that minimizes the total cost. To do this, we can take the derivative of the total cost with respect to v and set it equal to zero:
d(Total cost) / dv = 0
Differentiating the total cost equation:
(d(Cost of gas) / dv) - (300 * 20) / v^2 = 0
Now, we solve this equation to get the value of v.
Once we find the value of v, we can substitute it back into the total cost equation to calculate the minimum cost.
Finally, we can compare this minimum cost with the costs at speeds of 55 and 70 miles per hour by substituting those speeds into the total cost equation and calculating the costs. The speed that yields the lowest cost is the desired speed for the truck driver.
To find the speed at which the truck driver should be assigned in order to minimize the cost for the owner, we need to determine the speed at which the cost per mile is minimized.
Let's start by finding the cost function for the trip.
The cost of gas can be calculated by multiplying the rate of gas consumption by the cost per gallon. In this case, the rate of gas consumption is given as (1 + (1/400)*(v^2)) gallons per hour, and the cost per gallon is $2.60. Therefore, the cost of gas per hour can be expressed as (1 + (1/400)*(v^2)) * $2.60.
The time it takes to complete the 300-mile stretch can be calculated by dividing the distance by the speed, which is given as v miles per hour. Therefore, the time for the trip is 300/v hours.
To calculate the total cost for the trip, we multiply the cost per hour by the total time of the trip. The total cost function can be expressed as:
Total Cost = (1 + (1/400)*(v^2)) * $2.60 * (300/v)
To minimize the cost, we can differentiate the total cost function with respect to v and set it equal to zero. Then solve for v.
d(Total Cost) / dv = 0
Let's differentiate the equation:
d(Total Cost) / dv = (1/400) * (v^2) * (300/v^2) * $2.60 - 300/v^2 * $2.60
Simplifying the equation:
(1/400) * (v^2) * (300/v^2) * $2.60 = 300/v^2 * $2.60
Multiplying both sides by v^2 for cancellation:
(1/400) * (v^2) * 300 * $2.60 = 300 * $2.60
(1/400) * (v^2) * 300 = 300
Multiplying both sides by 400 for cancellation of (1/400):
v^2 * 300 = 300 * 400
v^2 = 400
Taking the square root of both sides:
v = √400
v = 20 mph
Therefore, the truck driver should be assigned to drive at a speed of 20 miles per hour to minimize the cost for the owner.
To compare the minimum cost to the cost when driving at 55 and 70 miles per hour, we can substitute these values into the Total Cost equation:
Cost at 55 mph:
Total Cost at 55 mph = (1 + (1/400)*(55^2)) * $2.60 * (300/55)
Cost at 70 mph:
Total Cost at 70 mph = (1 + (1/400)*(70^2)) * $2.60 * (300/70)
Calculate these costs for comparison.