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Best_Mathematician
Calculus challenge
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
once you find two x values , they are your answer for a)
is x two negative values
ya here is the equation again \[\frac{ x^2 }{ 4 } + y^2 = 1\]
ahh, that's different, this is ellipse x^2/4 + (y-3)^2 = 1
now plug in y=3, x^2/4=1 x^2=4 x=2,-2 that's your part a
sweet ur r doing grt go ahead
part b is graphing,
Here's the graph, done without benefit of the above
i wud just graph that right
oh thx @dlipson1 can u go further
Uh oh, I think that's a line integral, I'd have to look that up. Do you know anything about Stochastic Optimization?
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
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.
so from negative 2 to 2...right
Yeah, use the top half, y = sqrt(...), then (I just looked it up): Length = integral (sqrt(1+(y')^2))dy
ya and what kinda graphing calculator r u using dude
http://www.padowan.dk/ and Google --> http://tutorial.math.lamar.edu/Classes/CalcII/ArcLength.aspx
thx...getting my next question...i wud give u a lot of awards but unfortunately this site doesnt aloow lol
"Graph" (from padowan.dk) gives me the curve length of 4.882, then +3+3 +4 for the entire perimeter.
sweet thanks
hey @dlipson1 one more thing, how wud we find area on top of the toast
The rectangle + 2*integral (by symmetry) from 0 to 2 of y = sqrt(1-x^2/4)... hmm, do we need substitution here?
do u need the derivative...i have it
-x/(4y-12)....now wht to do
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
Here is almost the same problem: http://www.youtube.com/watch?v=PSlsj0IP8R8