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anonymous 3 years ago Calculus challenge

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1. anonymous

The type of bread chosen for this special calculus toast isn't the square sandwich shape, but the kind that is curved across the top. Imagine that the toast is composed of the curved part sitting atop the rectangular portion. The equation of the curved part of the toast is x2/4 + y2 = 1, and it sits directly and perfectly on top of a rectangle of height 3 inches. a) What are the equations of the rectangular boundaries? b) Graph the toast boundaries, making certain to include screen shots of the boundary equations, Window settings, and the graph. c) How would you find the length of the curve

2. anonymous

ok

3. anonymous

then

4. anonymous

once you find two x values , they are your answer for a)

5. anonymous

is x two negative values

6. anonymous

is x squred?

7. anonymous

ya here is the equation again $\frac{ x^2 }{ 4 } + y^2 = 1$

8. anonymous

ahh, that's different, this is ellipse x^2/4 + (y-3)^2 = 1

9. anonymous

now plug in y=3, x^2/4=1 x^2=4 x=2,-2 that's your part a

10. anonymous

sweet ur r doing grt go ahead

11. anonymous

part b is graphing,

12. anonymous

Here's the graph, done without benefit of the above

13. anonymous

i wud just graph that right

14. anonymous

oh thx @dlipson1 can u go further

15. anonymous

Uh oh, I think that's a line integral, I'd have to look that up. Do you know anything about Stochastic Optimization?

16. anonymous

hey hey hey never mind...i know how to find the length of the curve thanks..but i dont knw do i need to find length of whole curve or just the bread as in ur graph

17. anonymous

Well, the rectangle is trivial (is the side along the x-axis included?), the top is just half the ellipse, that's the only real calculus (integration) you have to do.

18. anonymous

so from negative 2 to 2...right

19. anonymous

Yeah, use the top half, y = sqrt(...), then (I just looked it up): Length = integral (sqrt(1+(y')^2))dy

20. anonymous

ya and what kinda graphing calculator r u using dude

21. anonymous
22. anonymous

thx...getting my next question...i wud give u a lot of awards but unfortunately this site doesnt aloow lol

23. anonymous

"Graph" (from padowan.dk) gives me the curve length of 4.882, then +3+3 +4 for the entire perimeter.

24. anonymous

sweet thanks

25. anonymous

hey @dlipson1 one more thing, how wud we find area on top of the toast

26. anonymous

The rectangle + 2*integral (by symmetry) from 0 to 2 of y = sqrt(1-x^2/4)... hmm, do we need substitution here?

27. anonymous

do u need the derivative...i have it

28. anonymous

-x/(4y-12)....now wht to do

29. anonymous

I'm not sure dy/dx helps. I set up the integral, thought about a trig. substitution, multiplied through by the 2 (as sqrt(4)) to get int(sqrt(4-x^2))dx, which I found here, but I think there must be an easier way (like polar coordinates): http://answers.yahoo.com/question/index?qid=20071231235113AAAAfuP

30. anonymous

Here is almost the same problem: http://www.youtube.com/watch?v=PSlsj0IP8R8

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