anonymous one year ago Guys please help me, I give medals If 1,200 cm2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic centimeters

1. anonymous

@Loser66

2. triciaal

|dw:1440636145968:dw|

3. triciaal

|dw:1440636232302:dw|

4. triciaal

volume changes with area and area changes with height

5. anonymous

okay, from there can I just solve for the h and then plug it into the volume function which is V=s^2*h

6. anonymous

|dw:1440636504171:dw|

7. triciaal

@jim_thompson5910

8. jim_thompson5910

solve for h to get $\Large 1200 = s^2 + 4sh$ $\Large 1200 - s^2 = 4sh$ $\Large 4sh = 1200 - s^2$ $\Large h = \frac{1200 - s^2}{4s}$

9. jim_thompson5910

$\Large V = s^2*h$ $\Large V = s^2*\frac{1200 - s^2}{4s}$ $\Large V = s*\frac{1200 - s^2}{4}$ $\Large V = \frac{1200s - s^3}{4}$ so it looks good so far

10. anonymous

the volume formula would be

11. anonymous

|dw:1440637090561:dw|

12. anonymous

right?

13. triciaal

yes read above by jim he found the expression for h then substitute in the formula for the volume

14. jim_thompson5910

yes correct, the volume is s^2*h

15. jim_thompson5910

you want to find the max volume, so you'll need to graph $\Large y = \frac{1200x-x^3}{4}$ and locate the local max

16. anonymous

What intervals should I use, My graph doesn't show it clearly.

17. jim_thompson5910

https://www.desmos.com/calculator/bbcpea5fld my window I used was xmin = -10 xmax = 40 ymin = -3000 ymax = 5000

18. jim_thompson5910

on the desmos graph, you should be able to click the local max (maybe click twice) to have the coordinates of that point show up

19. anonymous

is it 4000

20. jim_thompson5910

yeah that local max is (20,4000) meaning that x = 20 and y = 4000 so if the side length is s = 20 cm then the max volume is 4000 cubic cm

21. anonymous

OMG, thank you so much! ... DO you think you can help me with another one?

22. jim_thompson5910

sure

23. anonymous

provide one the two positive integers whose sum is 200 and whose product is a product

24. triciaal

what do you mean the product is a product?

25. anonymous

like if the two integers are multiplied, that equals to the maximun value

26. triciaal

the constant is not prime?

27. anonymous

I don't know to be honest

28. jim_thompson5910

do you mean provide one the two positive integers whose sum is 200 and whose product is a MAXIMUM ??

29. anonymous

yes

30. jim_thompson5910

integers whose sum is 200 x+y = 200 solve for y to get y = 200-x

31. jim_thompson5910

the product is the result of multiplying x*y = x*(200-x) = 200x - x^2

32. jim_thompson5910

the goal is to max out 200x - x^2

33. jim_thompson5910

you can graph 200x - x^2 or complete the square to find the vertex

34. anonymous

I got (100, 10000)

35. anonymous

@jim_thompson5910 so the answer is 10000

36. jim_thompson5910

so the two numbers are 100 and 100 100+100 = 200 100*100 = 10,000 the max product is indeed 10,000

37. jim_thompson5910

I think they are asking for one of the two numbers and not the max product

38. anonymous

okay, thank you

39. jim_thompson5910

sure thing

40. triciaal

@jim_thompson5910 thanks

41. jim_thompson5910

you're welcome