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A ship uses $( 7v^2 ) of fuel per hour when travelling at a constant speed of v km/h. Other expenses in operating the ship (i.e. crew and equipment) amount to $345 per hour. The ship makes a journey of 100 km.

C(v) : (700v^2 + 34500)/v
C'(v): (700v^2 - 34500)/v^2
C"(v): 69000/v^3

Question: To minimize the total cost of the journey the ship should travel at a speed of _____km/h
Correct to 3 significant figures.

(The speed that minimizes the total cost that have calculated correspond to a zero of the derivative of the total cost. The minimum of the total cost will (for this problem) occur at a turning point of the total cost function)


  • calculus -

    time spent is 100/v

    C(v) = 7v^2 * 100/v + 345 * 100/v
    = 700v + 34500/v (your C(v) is correct)

    However, your C' is way wrong.

    C'(v) = 700 - 34500/v^2

    set C'(v)=0, to get v=√(345/7) = 7.02 km/hr

    Better question would have asked for the speed in knots: 3.79

    Must be a barge!

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