## Azteck 2 years ago A truck is to be driven 200 km along a level highway at x km per hour. Petrol costs 8 cents per litre and is used at the rate of $25+\frac{ x^2 }{ 112 }$ litres per hour. The driver receives 2 dollars per hour. What is the most economical speed (to the nearest km per hour and the cost of the trip- i) if there is no speed limit, ii) if the speed must not exceed 60km per hour?

1. Azteck

Topic: Calculus Sub Topic: Maxima and Minima.

2. LogicalApple

Is it 74.83 km/h?

3. Azteck

Yes!!

4. LogicalApple

$C(x) = 0.08(25 + \frac{ x^{2} }{ 112 }) * \frac{ 200 }{ x } + 2(\frac{ 200 }{ x })$ This is the equation I came up with for Cost.

5. Azteck

75km/hour

6. Azteck

How did you do the first part?

7. LogicalApple

sorry

8. Azteck

gallon?

9. Azteck

American?

10. LogicalApple

let me rewrite that , (i used american units and accidentally wrote it in the wrong place)

11. LogicalApple

The first thing we know is that the rate of petrol is \$0.08/litre And the second thing we know is that the petrol is used at a rate of 25 + x^2/112 litre/h so $0.08 \frac{ dollar}{ litre } * (25 + \frac{ x^{2} }{ 112 }) \frac{ litre }{ hour }$ Gives us the units dollars/hour

12. LogicalApple

The only thing we need to know is how long the trip takes. We know speed = distance / time speed = x distance = 200 km time = t Therefore time, t = 200/x If we multiply what we have so far by (200/x) (which is in hours), then the hours unit cancels leaving an expression in dollars

13. Azteck

So far, I'm up to: $0.08*(25+\frac{ x^2 }{ 112 })$

14. LogicalApple

$0.08 \frac{ dollar }{ litre } * (25 + \frac{ x^{2} }{ 112 }) \frac{ litre }{ hour } * \frac{ 200 }{ x } hours$

15. Azteck

So you get that, and do you then add: $\frac{ 400 }{ x }$ for the money he receives per hour?

16. LogicalApple

yes

17. Azteck

okay. Now I'm getting somewhere.

18. Azteck

So do we differentiate that with respect to x?

19. LogicalApple

Yes because C(x) is the cost evaluated at x. So we need to minimize cost by differentiating with respect to x.

20. Azteck

Thanks for the assistance @LogicalApple . Appreciate it.

21. LogicalApple

Not a problem

22. LogicalApple

For the second part of the question, keep in mind that the point we evaluated in the first part was a minimum. Due to the nature of the hyperbola, you can consider it an absolute minimum whenever x > 0 This should help you answer the second part of the question.

23. Azteck

Yeah I already finished the question.

24. LogicalApple

Oh, nice

25. Azteck

The hard part was what you did for me.

26. LogicalApple

I did it on paper and I was like "hm.. I can't think in litres/gallon"

27. LogicalApple

Ah, I did it again lol I mean litres/hour!

28. Azteck

hahah.

29. Azteck

Well I'm going to close this and put up another question. Thanks again for helping me.

30. Azteck

Hope you can help me again if you see fit to.

31. LogicalApple

Good luck!