## Best_Mathematician 2 years ago Calculus challenge

1. Best_Mathematician

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. Best_Mathematician

ok

3. Best_Mathematician

then

4. modphysnoob

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

5. Best_Mathematician

is x two negative values

6. modphysnoob

is x squred?

7. Best_Mathematician

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

8. modphysnoob

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

9. modphysnoob

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

10. Best_Mathematician

sweet ur r doing grt go ahead

11. modphysnoob

part b is graphing,

12. dlipson1

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

13. Best_Mathematician

i wud just graph that right

14. Best_Mathematician

oh thx @dlipson1 can u go further

15. dlipson1

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

16. Best_Mathematician

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. dlipson1

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. Best_Mathematician

so from negative 2 to 2...right

19. dlipson1

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

20. Best_Mathematician

ya and what kinda graphing calculator r u using dude

21. dlipson1
22. Best_Mathematician

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

23. dlipson1

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

24. Best_Mathematician

sweet thanks

25. Best_Mathematician

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

26. dlipson1

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

27. Best_Mathematician

do u need the derivative...i have it

28. Best_Mathematician

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

29. dlipson1

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. dlipson1

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