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kirill
1 month ago
11

The fuel cost per hour for running a ship is approximately one half the cube of the speed (measured in knots) plus additional fi

xed 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
Mathematics
1 answer:
Zina [12.3K]1 month ago
7 0

Answer:

6 knots

Step-by-step explanation:

Let the velocity be v knots

thus, the time required to traverse 500 M is given by 500 / v hours

Fuel consumption per hour is equal to 216 plus half the cube of the speed (v^3).

Let F denote the fuel consumption for the journey

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

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

The derivative of F with respect to v is: dF/dv = 500[ - 216/v^2 + v]

The second derivative, d^2F/d^2v = 500[432/v^3 + 1], indicates positivity.

Setting dF/dv to zero helps find the minimum.

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

or v = 216/v^2

or v^3 = 216

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

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