Question

The cost of fuel per kilometre for a truck travelling v kilometres per hour is given by the equation:

C(v)=100v​+v/25​.

a) What speed will result in the lowest fuel cost per kilometre?
b) Assume the driver is paid $40/h. What speed would give the lowest cost, including fuel and wages, for a 1000−km trip?

Answer 1

To find the speed that results in the lowest fuel cost per kilometre, we need to find the minimum value of the function C(v) = 100v + v/25. Let's find the derivative of C(v) first: C'(v) = 100 + 1/25. Setting C'(v) equal to zero, we find that there is no specific speed that results in the lowest fuel cost per kilometre. To find the speed that gives the lowest cost, including fuel and wages, for a 1000-km trip with a driver paid $40/h, we need to calculate the cost function and minimize it.

Step-by-step explanation:

To find the speed that results in the lowest fuel cost per kilometre, we need to find the minimum value of the function C(v) = 100v + v/25. This can be done by taking the derivative of the function and setting it equal to zero. Let's find the derivative of C(v) first:

C'(v) = 100 + 1/25

Setting C'(v) equal to zero:

100 + 1/25 = 0

100 = -1/25

There is no solution to this equation, which means the function does not have a minimum value. Therefore, there is no specific speed that results in the lowest fuel cost per kilometre.

To find the speed that gives the lowest cost, including fuel and wages, for a 1000-km trip with a driver paid $40/h, we need to calculate the cost function for a given speed and minimize it. The cost function, inclusive of fuel and wages, can be defined as:

C(v) = 100v + v/25 + (40 * 1000 / v)

By taking the derivative of the cost function and setting it equal to zero, we can find the speed that results in the lowest cost.