## anonymous one year ago a ladder 6 m long leans against a vertical wall. The lower end of the ladder is moved away from he wall at the rate of 2 m/min. Find the rate of change of the area formed by the wall, the floor and the ladder when the lower end is 4 m from the wall. please help

1. anonymous

|dw:1443859889821:dw|

2. anonymous

It doesnt seem right for some reason

3. IrishBoy123

can you think of an equation that connects l and b in your drawing?

4. anonymous

|dw:1443860176242:dw|

5. anonymous

A=1/2 bh

6. anonymous

Using the pythagorean theorem, we can find the rate of change for everything moving

7. dan815

oh boy a ladder problem :)

8. anonymous

what is the area if the base?

9. anonymous

is 4

10. anonymous

You don't necessarily need to know that. You're simply looking for $$\dfrac{dl}{dt}$$ when $$x=4$$

11. anonymous

how?

12. IrishBoy123

Pythagoreas

13. anonymous

The area is constantly changing w.r.t time as the ladder moves down the wall

14. dan815

This question you are doing right now, will make or break your mathematical career

15. anonymous

Dan... -_-

16. anonymous

there's no change in y right? the ladder?

17. Astrophysics

|dw:1443860459057:dw| $\frac{ dx }{ dt } = 2$ I thought this

18. anonymous

so i'll just substitute dy/dt and dx/dt?

19. anonymous

to find dl/dt?

20. dan815

|dw:1443860606641:dw|

21. anonymous

Yeah you're right, it moved away from the wall at that rate, whoops.

22. Astrophysics

:)

23. anonymous

y=2 sqrt of 5?

24. dan815

yes that is sqrt(2)

25. dan815

sqrt(20)*

26. dan815

but solve for dy/dt

27. Astrophysics

yes dy/dt is what you need

28. anonymous

I think i read the question wrong, fml.

29. dan815

|dw:1443861020387:dw|

30. anonymous

Oh okay, as the ladder moves down the wall, the ladder increases the base of the triangle but shortens the height of the triangle, therefore you're looking for the rate of change of the height

31. Astrophysics

$x^2+y^2=36$ $2x \frac{ dx }{ dt }+2y \frac{ dy }{ dt }=0$ $\frac{ dy }{ dt }=-\frac{ x }{ y }\frac{ dx }{ dt }$

32. Astrophysics

So x here is the distance from the bottom of the ladder to the wall and y is the distance of the top of the ladder to the ground, as the image shows. The second part of the equation we simply use chain rule, $2x \frac{ dx }{ dt }+2y \frac{ dy }{ dt }=0$ and we just solve for dy/dt then

33. anonymous

wait is this right? dh/dt =-2sqrt5 / 5 that means the Area now is 6/sqrt5?

34. Astrophysics

The rate should be negative because the ladder is decreasing

35. Astrophysics

it just defines the direction

36. anonymous

or dy/dt= -2/sqrt5

37. Astrophysics

Well what did Pythagorean theorem give you? x=4, r=6 $x^2+y^2=r^2 \implies y=\sqrt{36-16} = \sqrt{20}$

38. Astrophysics

And we know $\frac{ dy }{ dt }=-\frac{ x }{ y }\frac{ dx }{ dt }$ just plugging in the values at this point

39. Astrophysics

Yes dy/dt = - 2/sqrt(5)