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.
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.