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Chapter 2 : Scaling and Measurement
Study Questions and Projects
- Find the dimensions of the Great Pyramid, and some of the lesser ones.
If we double the length of the base of a pyramid, how does that change the
height? What about the amount of plaster on the outside? What about the
total number of blocks?
- In doubling the width of a 3' by 6' quilt with a 4" border, how will
the area of the border change? If there are 200 squares in the inner part,
how many with there be in the full-sized quilt? How much longer does the
border become? What if we double both the width and the length?
- A pylon is an example of a pyramid where the height is much larger than
the lengths of the sides of the square base. How do the surface area and
the volume change if the height is kept the same and the base is made into
a square with twice the side length?
- If we double the size of a triangular pyramid, the volume goes up by a
factor of eight. If we cut off a pyramid from each corner of a pyramid,
going down to the halfway point, what is the nature of the figure that is
left? What is its volume?
- What if we truncate a pyramid by going one-third of the way in? What is
the nature of the figure and what is it's volume?
- What if we cut away four pyramids from the corners of a cube? What is
left, and what is its volume?
- A baby 20" and 9 lb. at birth would weigh how much if it grew to a
height of five feet and kept the same proportions? What proportions change
most dramatically as a person triples his or her height?
- In the cube with each side divided into three parts, how many cube have
three sides showing? How many have two? One? Zero?
- What if the cube is divided into four parts? Into n parts?
- How many segments are there in a subdivided square? How many in a
subdivided cube?
- If we double the radius of a disc, what happens to its diameter? Its
circumference? Its area?
- What if we double the height of a circular cylinder keeping its radius
fixed? What if we double the radius keeping the height fixed? If we
multiply the radius by some factor k, what do we have to do with the
height in order to maintain the same volume?
- Of all pyramids based on the same square and with the same height,
which has the smallest surface area?
- What about the same question for a pyramid with base given by a regular
polygon? Give an argument to show that the volume is
1/3 area of base times height, assuming that this is
true for a triangular pyramid.
- Why should it be true in general that the volume of a cone with a fixed
base should be 1/3 area times height?
- Same question for prisms.
- What is the formula for the volume of a truncated cone, in terms of the
height, upper radius, and lower radius? How does this compare to the
Egyptian formula for the volume of a truncated pyramid?
- An octahedron can be considered as eight triangle-based pyramids, each
with volume 1/6 that of a cube. What is the analogue
of this statement in the second dimension? In the fourth?
- What happens to the volume of the n-cube as n goes to
infinity? What about the volume of the n-dimensional octahedron?
Which octahedron has the largest volume?
- Which octahedron has the largest generalized area?
- Make a model of an unfolded off-center pyramid, and compute its surface
area. What is the analogous computation for a hyperpyramid?
- Show how to intepret a2 - bi>2 as the area of two
rectangles, each with one side of length a - b. Generalize to three-dimensional
space.
- What can we say about the areas of equilateral triangles erected on the
sides of a right triangle? Is there a nice geometric proof of the result in
this case? What about semicircular areas?
- What if we remove the middle cube of a subdivided cube, and then do
this over and over again? What will the the volume of the figure that is
left?
- What if we subdivide a cube into thirds and keep only the corners, then
do this over and over again? What is the volume of the remaining figure?
How does this compare with the triangular Sierpinski gasket?