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I was studying for some exams when I tripped over this question:

The cost of fuel per hour for running a ship is proportional to the cube of the speed and is \$27 per hour when the speed is 12 miles per hour. Other costs amount to $128 per hour regardless of the speed. Express the cost per mile as a function of the speed, and find the speed that makes this cost a minimum.

In my book, the answer is $$C = \frac{v^2}{64} + \frac{128}{v}.$$

I don't know how the book came up with that equation. I wanna be enlightened.

I prefer to show the units being cancelled so that the cost $C$ is in dollars per mile:-)

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What they are telling is that $$C=k v^3+128$$ and that $$k 12^3=27 \implies k=\frac{27}{12^3}=\frac{27}{1728}=\frac{1}{64}$$ This is the cost per hour. But, in one hour, the ship run $v$ miles. So the cost per mile is just $$\frac C v=\frac{\frac{v^3}{64}+128} v=\frac{v^2}{64}+\frac {128}v$$ and this is what you want to minimize.

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