Geometry for Teachers

[cabriosnob]

Basic paramters of problem: 10in x 10in cake, 2in tall. Will have top and sides frosted. 10 people sharing who need equal amounts of cake and frosting, then I have to sketch on graph paper how the pieces will be cut. Then, steps for cutting square cake size (s) into (n) equal pieces of cake and frosting. We are told that the pieces of cake will be measured in volume and the frosting measured in area covered by frosting.

So far, I don't have much. Please tell me where I've gone off course/am missing something...

I have the volume of the cake as 200in [sup:2yhfdgxn]3[/sup:2yhfdgxn] and figured surface area for frosting, but not using 2B because the bottom will not be frosted. So, I used
SA=B + ph and came up with 180in.

To split that between 10 people means each person will get 20in[sup:2yhfdgxn]3[/sup:2yhfdgxn] of cake and 18 in frosting? Not only dies that sound wrong to me, I have no idea how to sketch it. If I cut the middle pieces bigger to compensate for the extra frosting on the side pieces, that will change the volume of those pieces making them all unequal. I considered cutting more than 10 pieces and maybe everyone getting a side and middle piece, but can't get my head around it.

You guys helped so much with the last problem solving report, I was hoping for some guidance on this one. I'm really stuck. This is for a class on teaching Geometry, focusing on elementary school.

 

[cabriosnob]

Can I cut the middle pieces in a rectangular shape and the side pieces in square shapes? That would maintain equal volume, right?

 

[stapel]

Hint: The area A of a triangle with height h and base b is given by A = (1/2)bh. So any triangular piece, cut from the center to the sides and having the same length of "edge" as its base, must necessarily have the same area. Since the cake is of uniform thickness, such pieces must necessarily have the same volume. :wink:

Eliz.

 

[cabriosnob]

Thanks! I feel so dumb!!! I went through everything but triangles!!! I knew I was making this way harder than it has to be...Thank you so much for the help!!!