Question

Suppose the cost per hour incurred in operating a cruise ship is 3a + bv3 dollars per hour, where a and b are positive constants and v is the ship's speed in miles per hour. At what speed (in miles per hour) should the ship be operated between two ports, at a distance D miles apart, to minimize the cost? (Hint: Minimize the cost, not the cost per hour.)

Answer 1

Answer:

Suppose the cost per hour incurred in operating a cruise ship is 3a + b $v^{2}$dollars per hour, where a and b are positive constants and v is the ship's speed in miles per hour. At what speed (in miles per hour) should the ship be operated between two ports, at a distance D miles apart, to minimize the cost? (Hint: Minimize the cost, not the cost per hour.)

The speed at which the ship would maximize cost is $\sqrt[3]{\frac{3a}{2b} }$

Explanation:

The problem can be solved using differentiation to get the minimum value of the speed to travel between the two ports. Step by step calculation is contained in the attached images;