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27 March, 03:32

The fuel cost per hour for running a ship is approximately one half the cube of the speed (measured in knots) plus additional fixed costs of $216 per hour. Find the most economical speed to run the ship for a 500 M (nautical mile) trip. Note: Assume that there are no major disturbances, such as heavy tides or stormy seas

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  1. 27 March, 03:56
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    6 knots

    Step-by-step explanation:

    Let the speed be v knots

    then time taken to cover 500 M = 500 / v hrs

    fuel consumption / hr = 216 + 0.5v^3

    let F be the fuel consumption for trip

    = [500/v][216 + 0.5v^3]

    = 500[216/v + 0.5v^2]

    dF/dv = 500[ - 216/v^2 + v]

    d^2F/d^2v = 500[432/v^3 + 1], i. e. + ve

    so setting dF/dv will give a minima

    500[ - 216/v^2 + v] = 0

    or v = 216/v^2

    or v^3 = 216

    solving, we get v = [216]^ (1/3) = 6 knots
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