1920 Calculus Horror

Calculus Level 4

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

Anshuman Singh Bais by Anshuman Singh Bais , 21, India

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1 solution

Anubhav Tyagi
Dec 23, 2016

Tons of fuel consumption per hour = 0.3 + 0.001 v 3 + 1 = 1.3 + 0.001 v 3 The total distance of journey is ’d = 1000 nautical miles’. Total time of journey = 1000 v h r s Total fuel consumed during journey = 1000 v × ( 1.3 + 0.001 v 3 ) = 1300 v + v 2 Cost of journey C(v) = 10 × ( 1300 v + v 2 ) = 13000 v + 10 v 2 For minimum cost C’(v) = 0 ⇒ − 13000 v 2 + 20 v = 0 ⇒ v 3 = 650 v = 8.66239 nautical miles per hour To check if v=8.66 is minima, C”(v) > 0 C ′ ′ ( v ) = 26000 v 3 + 20 ⇒ C ′ ′ ( 65 0 1 3 ) = 60 > 0 Hence v = 8.66239 is the mimima \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}

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@Calvin Lin , @Pi Han Goh , - Check the solution. Also level down the problem level. Its over rated

Anubhav Tyagi - 4 years, 5 months ago

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Set to lvl 3. Your solution is almost complete. To prove that it's minimum, you need to show that C''(cube root of 650) is positive.

Pi Han Goh - 4 years, 5 months ago

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I think that should be a homework for the solvers or do you think should I do that as well?

Anubhav Tyagi - 4 years, 5 months ago

@Anshuman Bais It should be made clear in the term itself, that we are calculating the hourly consumption , and not the implied total consumption.

Calvin Lin Staff - 4 years, 5 months ago

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