## mathcalculus 2 years ago An orchard contains 51 peach trees with each tree yielding an average of 51 peaches. For each 4 additional trees planted, the average yield per tree decreases by 3 peaches. How many trees should be planted to maximize the total yield of the orchard? The number of trees is . (Give your answer as a whole number.)

1. mathcalculus

I got my answer 17/.. but they want it as a whole number. any ideas what i might have done wrong?

2. mathcalculus

@agent0smith

3. mathcalculus

@pgpilot326

4. mathcalculus

@zepdrix

5. mathcalculus

17/2*

6. pgpilot326

total peaches = (51 - (3/4) x)*(51+x) where x = # number of additional trees. maximize and if it's not an integer, which ever integer is closest will do the job.

7. mathcalculus

yes i did it exactly like that

8. pgpilot326

and the max occured at 17/2?

9. mathcalculus

then i multiplied them out and to maximize I found the derivative.

10. mathcalculus

well i got f(x)= -3x^2/4+51x/4+2601

11. mathcalculus

and f '(x)= -6x/4+51/4

12. mathcalculus

i set that to zero and got 17/2

13. pgpilot326

so check the yield at x = 8 and x = 9. they're likely the same so either would work. but if you're a farmer, you go with less additional trees to save \$

14. pgpilot326

by yield, i mean total yield, not yield per tree.

15. pgpilot326

x = 8 => 2655 peaches x = 9 => 2655 peaches

16. mathcalculus

i tried those also

17. mathcalculus

it says wrong

18. mathcalculus

19. pgpilot326

did you try 8?

20. mathcalculus

yeah

21. mathcalculus

:/

22. pgpilot326

i don't know what to tell you. could it be they want 59 as the asnwer (total number of trees in the orchard)?

23. mathcalculus

woahhhh how is it 59?

24. mathcalculus

its saying correct to that one

25. mathcalculus

did you use the second derivative test? or something?

26. pgpilot326

51 + 8 = 59. there were 51 already planted plus the 8 you added to maximize the total yield.

27. mathcalculus

but if it's 8.5 shouldnt we round it to 9?

28. mathcalculus

so wouldnt it be 60?

29. pgpilot326

question wasn't clear... should have said total number of trees planted in the orchard. you have to think like a farmer. one less tree gives the same yield as one more. what would be better? think of not only the initial cost of the tree but also the watering and care, etc.

30. mathcalculus

ok...

31. pgpilot326

maybe 60 would also work (as an answer) but it's not as practical as 59.

32. mathcalculus

im a little confused on that.

33. mathcalculus

yeah i tried 60 but it kept saying wrong.

34. mathcalculus

i mean mathematically speaking best round is 60 and then adding it to the already planted trees

35. mathcalculus

meaning 51+ 9....

36. pgpilot326

again, you have think like a business person/farmer. think of the problem like this: if you had to cut off fingers to get gold coins in the same scenario, you'd want to cut off as few fingers as possible in order to maximize the number of gold coins. if you got the same number of gold coins for cutting of 8 fingers as you would for cutting off 9 fingers, how many fingers would you cut off?

37. mathcalculus

huh?

38. pgpilot326

how old are you?

39. mathcalculus

this example sounds scary.

40. pgpilot326

yeah, intentionally.

41. mathcalculus

i understand you, but i just don't like the example.lol

42. mathcalculus

thanks anyways, i see the problem clearly now.

43. pgpilot326

lol, sorry. it's just what popped out of my head.

44. mathcalculus

thanks again!

45. pgpilot326

excellent. ask your teacher to be more explicit in what is wanted next time. you're welcome!