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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lawrence2top lawrence2top · Answer 1 · 2020-04-16T03:51:45+02:00

Answer:

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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andromache andromache · Answer 2 · 2022-03-25T14:08:05+01:00

The most economical speed to run the ship for a 500 M (nautical mile) trip should be 6 knots.

here we assume speed be v knots

So, time taken to cover 500 M = 500 / v hrs

Now

fuel consumption /hr = [tex]216 + 0.5v^3[/tex]

Also, let us assume F be the fuel consumption for trip

So,

[tex]= [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]\\\\[/tex] , i.e. +ve

Now

[tex]500[ -216/v^2 + v] = 0\\\\v = 216/v^2\\\\v^3 = 216[/tex]

v = 6 knots

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