2. The floor to be carpeted is an area and hence has a dimension of 2.
Increasing the linear dimensions of the room by a factor of 5 will
increase the area by a factor of 5^2 = 25. Thus, the total cost
of the carpet for the bigger room will be 25 x $130 =
$3250.
3. There are 100 centimeters in a meter. A cubic meter is a
measure of volume, so the dimension is 3. As a result, there
are 100^3 = 1,000,000 cubic centimeters in a cubic meter.
4. The beaker is holding a volume of water, so the dimension here is
3. Scaling up the beaker by a factor of 3 will scale up the
volume by a factor of 3^3 = 27. Thus, the big beaker can hold
27 x .75 liters = 20.25 liters.
5. Don't feel bad if you had trouble with this problem. I think that
this example is a good bit harder than the snowflake example we
did in class.
Notice that the big triangle appears to be made up of 3 smaller, identical triangles. Thus, three small copies of the triangle fit inside one big triangle. But what's the scale factor? Notice that 2 bases of the little triangles fit on the base of the large triangle. So the scale factor here is 2; all sides grow by a factor of 2. So,
3 = 2^D.
Still confused about the dimension of the Sierpinski triangle? Check out this excellent web page on Fractal Dimension by Bob Devaney.