## MarcLeclair 2 years ago A sherman is in a rowboat on a lake and 3 km from shore. He wishes to reach a store 2 km down the (straight) shore. He can row at 5 km/h and run at 13 km/h. To what point down-shore should he row to get to the store as quickly as possible? It's supposed to be optimization but I'm wondering if I need to use related rates... :/

1. completeidiot

did you draw a picture?

2. completeidiot

i dont think you need to use related rates for this question

3. MarcLeclair

Yup gives me a right angle triangle |dw:1354488913281:dw|

4. MarcLeclair

the store being the point where 5 km/h and 13km/h meet

5. completeidiot

you drew it wrong

6. completeidiot

you cant use speed as distance values

7. MarcLeclair

im just showing the distance where the speed is going. I thought it was normal to do. ( well for me to understand)

8. completeidiot

|dw:1354489054440:dw|

9. completeidiot

|dw:1354489176738:dw|

10. MarcLeclair

Oh yeah ended the triangle too late. I need to find the the point where he reaches the shore, But I usually get 2 equations, ( one is my constraint the other is the one I'm trying to maximize/minimize) I can't figure them out here Perimeter, or area, I mean I feel as if I don't have enough variables.

11. completeidiot

d/v=t |dw:1354489441544:dw| first determine the rowing distance $d_{row}=\sqrt{x^2+9}$ and running distance $d_{run}=2-x$ now calculate total time to travel said distance $t= d_{row}/v_{row} + d_{run}/v_{run}$ you should get a nice fun equation then solve for x using first derivative then solve for t

12. MarcLeclair

Ahhhhhh so its not always with area and peremeter.... stupid me. THanks a lot , I'll try it from there on my own :P Gotta love math

13. Chlorophyll

@MarcLeclair Have you found x yet?

14. MarcLeclair

No still trying hehe. Taking some time :(

15. Chlorophyll

Total time T = run time + row time ( 2- x ) / 13 + √ ( x² + 9 ) / 5 Can you take derivative:

16. MarcLeclair

yeah thats what im trying to do right now, I got (-13x/13 ^2) +3(1/2( x ^2+9) ^(-1/2) (2x) / 9

17. MarcLeclair

er I dont mean 3 I mean 5 sorry got confuse in the numbers. so 5(1/2 * and the last one is /15

18. Chlorophyll

T' = x / 5√ ( x² + 9 ) - 1 / 13

19. MarcLeclair

Isn't mine right but not simplified?

20. Chlorophyll

T' = 0 x / 5√ ( x² + 9 ) = 1 / 13 5√ ( x² + 9 ) = 13x

21. MarcLeclair

Yes thank you chlorophyll :) I ll verify if my derivative was right. Thanks a lot for your help

22. MarcLeclair

It is pretty complicated, I've got more complicated to do haha. I'm enjoying it O_o

23. Chlorophyll

13x = 5√ ( x² + 9 ) -> 169x² = 25 ( x² + 9) 144x² = 225 x = 15/12

24. MarcLeclair

That's what I got too and it is right. You two were life saviour haha

25. Chlorophyll

=> x = 1hr 15'

26. MarcLeclair

its okay hahahaha, everybody have their strong points, you still helped with calculations :) I'm aweful at making relationships in math :P

27. Chlorophyll

Finally, plug x to find the distance, though :) All credit should go to @completeidiot =)