Assignment 1 Math 355 Fall 2007
Due: 12N Friday, 21 Sept
Writing
•Using concise and clear sentences, incorporate symbols, calculations, and, above all, illustra-
tions into the text. Have an audience in mind. Explain what you’re doing and why you’re
doing it.
•You may consult classmates and the instructor, but what you write should be entirely your
own work.
•Typed work is appreciated. Hand-drawn figures are acceptable.
1) You want to slice a spherical orange some number of times so that you get the maximum
number of pieces. We know what happens if you can slice the orange with complete free-
dom. What happens if each slice passes through the oranges’s center? Determine the
appropriate rules for slicing and then develop a formula that solves the problem.
2) Suppose you stack blocks so that they form a square-based pyramid. On each layer of the
pyramid, you place a square of blocks. For instance, if the pyramid has three layers, the
bottom layer has 9 = 32blocks, the second layer has 4 = 22blocks, and the top layer has 1
block.
a) Determine the total number of blocks used in an n-layered pyramid. Prove your claim.
b) Use the result to determine the volume of a smooth square-based pyramid (where the
base is a×aand the height is h). Suggestion: Compare the pyramid to a block that’s
a×a×h.
3) Picture proofs: What formula does the picture “prove?” Describe how to “see” the formula
in the picture.
a) b)
