__Category__: Differential Calculus, Algebra"Published in Newark, California, USA"

The cost of fuel per hour for running a ship is proportional to the cube of the speed and is $27 per hour when the speed is 12 miles per hour. Other costs amount to $128 per hour regardless of the speed. Express the cost per mile as a function of the speed, and find the speed that makes this cost a minimum.

__Solution__:

The first thing that we have to do is to analyze the given word problem as follows

Let C = cost of fuel per hour

v = speed of a ship

From the word problem, "The cost of fuel per hour for running a ship is proportional to the cube of the speed", the working equation will be

If C = $27 per hour

v = 12 miles per hour

then the value of k will be

Therefore

From the word problem, "Other costs amount to $128 per hour regardless of the speed", the final working equation will be

To get the minimum cost of a fuel as a function of the speed, take the derivative of the above equation with respect to v as follows

Set

^{dC}/

_{dv}= 0 since we are getting the minimum cost of a fuel as follows