Probem37+Geneveive+Ben+Z

A cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup.

Genevieve:So here's what we know. The volume of a cone is: V=1/3 πr^2h  Maximize Volume. So, if you could see the diagram, it would show a circle with a sector cut out of it. If you folded this circle up, it would create this cone. If we took a direct side view of the cone, it would be a right tringle. h being the vertical line, R being the hypotenuse, and little r being the horizonal line. We could use Pythagorean theorem to say that R^2=h^2 + r^2. Using the side view triangle, we could solve for the little r using Pythagorean theorem. So it would equal r^2= R^2 - h^2. We could plug "that" into the volume equation of a cone. That would look like this: V= 1/3 π(R^2 -h^2)h V= 1/3 π(hR^2 -h^3) Then, we take the derivative, because we want to maximize the volume, and where the slope is 0, that's where it will be at it's maximum volume. dV/dh= 1/3π(dhR^2 - 3h^2dh) Set this equal to 0 and simplify. 0=(dhR^2 - 3h^2dh) //(ms sweeney: get rid of the 1/3π here!)//

Genevieve: 3h^2dh=dhR^2

Now, divide by dh and simplify: h=sqrt( 1/3*R^2) Now, we know the maximum volume occurs at this HEIGHT! So, plug this into the volume equation: So, the final answer is: V= 1/3 π*sqrt(r^2/3)*(R^2 -r^2/3) (Genevieve: this is the revised volume equation, ms. sweeney!) Ok, so do you think this seems right, ben?

Ok sorry for coming on so late, I don't really know how to make a new page and all that so let me look over all of this.... Well h can really be any positive number so that means we should find the limit  as it approaches 0 and as it approaches infinity. Also thinking about how slant height is related to h and r, I think it forms a triangle. So it would be h^2+R^2=l^2? Ok I'm trying to visualize this, but I'm a little confused. Would the endpoints be infinity and 0? Or you can say that they are limited somehow by the slant height and express it in different terms?