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It costs 32000+2v^3 dollars to operate a cruise ship for an hour, where v is the velocity of the ship measured in km/hr. How long will it take for a trip over 120 km if the total cost is to be minimized.

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The cost to operate the cruise ship for one hour is given by C = 32000 + 2*v^3. At a velocity of v, the distance covered by the ship in an hour is v km.

Let the optimal velocity for the trip of 120 km be V. The time taken to cover 120 km at this speed is 120/V. The cost involved is (32000 + 2*V^3)*(120/V)

C = 32000*120/V + 2*V^3*120/V

=> C = 32000*120/V + 240V^2

To minimize the cost we have to find the minimum value of C.

This is obtained by solving for C' = 0

=> -32000*120/V^2 + 480V = 0

=> 32000*120/V^2 = 480V

=> V^3 = 32000*120/480

=> V^3 = 8000

=> V = 20

At this speed the time required to complete the trip is 120/20 = 6 hr.

If the total cost is to be minimized it would take 6 hours to complete the trip.

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