The consumption of coal by the 1920's steamer
*
Ayumo
*
may be represented by the formula
$y=0.3+0.001v^{3}$
, where
$y$
represents number of tons of coal burned per hour and
$v$
represents speed in nautical miles per hour.

Other costs of maintenance of ship and wages, per hour, is same as cost 1 ton of coal.

What speed (in nautical mules per hour) will minimize the total cost of a voyage of 1000 nautical miles?

**
Bonus:
**
What will be that minimum cost of voyage if 1 ton of coal = 10 shillings

The answer is 8.66.

**
This section requires Javascript.
**

You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.

$\begin{aligned} &\text{Tons of fuel consumption per hour } = 0.3 + 0.001v^3 +1 = 1.3 + 0.001v^3 \\ &\text{The total distance of journey is 'd = 1000 nautical miles'. Total time of journey } = \frac{1000}{v} hrs \\ &\text{Total fuel consumed during journey = } \frac{1000}{v} \times (1.3 + 0.001v^3) = \frac{1300}{v} + v^2 \\ &\text{Cost of journey C(v) } = 10\times \Big(\frac{1300}{v} + v^2\Big) = \frac{13000}{v} + 10v^2 \\ &\text{For minimum cost C'(v) = 0}\\ &\Rightarrow \frac{-13000}{v^2} +20 v = 0 \\ &\Rightarrow v^3 = 650 \\ &v = 8.66239 \text{ nautical miles per hour}\\ &\text{To check if v=8.66 is minima, C''(v) > 0 } \\ &C''(v) = \frac{26000}{v^3} +20 \Rightarrow C''(650^{\frac{1}{3}}) = 60 >0 \\ &\text{Hence v = 8.66239 is the mimima} \\ \end{aligned}$