Problem #1.
1. A figure composed of congruent
cubic blocks has four layers. On the lowest layer are 7 rows of 7 blocks each.
Centered on the bottom layer are 5 rows of 5 blocks each. Centered
on top of that are 3 rows of 3 blocks each. Finally, a central block is
placed on top of the entire structure. See the picture below. After the blocks
are placed, the entire figure is spray painted from the sides and top. How many
blocks have 6 painted sides, how many have 5, 4, 3, 2, 1 and 0 painted sides?
Suppose a similar figure has 99 rows of 99 blocks each on the bottom layer. How
many have 6, 5, 4, 3, 2, 1 and 0 painted if the structure is built similar to
the one shown?
Most students approached this from an analytical basis. Starting with those that have no color
0 painted sides – a pyramid
with 5, 3, and 1 cubes or 25 + 9 +1 = 35
1 painted side - none of the
cubes have one painted side = 0
2 painted sides – All the
cubes along the edge and on the surface 4x5 + 4x3 + 4x1 = 36
3 painted sides – All the
corner surface cubes = 12
4 painted sides – None
= 0
5 painted sides – The top
cube = 1
6 painted sides – None
= 0
Total cubes = 35 + 0 +
36 + 12 + 0 + 1 + 0 = 84 = 49 + 25 + 9 + 1
Problem # 2
2. Suppose that a regular octagon
is filled as shown in the figure below. What is the area of the shaded region
if each side of the octagon is 4 cm long?
The two right triangles that make up the shape
have legs of 4 and (4 + 42). Because there are two similar triangles, the total area would be 2 * 1/2
b * h or 2 * ½ *
(4 + 42) * 4 or
approximately 90.51
Problem #3
3. There are 10 lines in a plane
each intersecting the other nine. No three lines pass through the same point.
At how many points do the lines intersect, and into how many regions do they
separate the plane?
Each is a mathematical progression shown in the table below
|
Lines |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
Regions |
2 |
4 |
7 |
11 |
16 |
22 |
29 |
37 |
46 |
56 |
|
Intersections |
0 |
1 |
3 |
6 |
10 |
15 |
21 |
28 |
36 |
45 |
The formula for finding the number of regions (given n lines) is: 0.5n2
+ 0.5n + 1
The formula for finding the number of intersections (given n lines) is: 0.5n2 – 0.5n
Problem #4
4. Counting only paths that follow
the lines and go downward or to the right, how many paths for A to D are
possible?
Problem #5
5. A six pointed regular star
consists of two interlocking equilateral triangles. What is the ratio of the
area of the entire star to the area of one of the overlapping equilateral
triangles?
The six pointed star is comprised of 12 small equilateral
triangles. One of the two equilateral
triangles is comprised of 9 of the small ones.
Therefore the ratio of the entire star to one of the large equilateral
triangles is 12/9 ot 4/3 or
1.3333…
Problem #6
6. Points A, B, C, and D lie on the same line, in the order written. If AB:AC = 1:3 and BC:CD = 4:1, compute the ratio AB:CD.
The following number line meets the requirements. You will notice that AB:CD
is a ratio of 2 to 1.
Problem #7
7. Two identical oil pipelines are circular cylinders of radius 6". The Environmental Protection Agency has ordered that they be replaced with a single pipeline with the same capacity. If the new pipeline is also a circular cylinder, what must the radius be?
The capacity of a pipe is measured by its
cross sectional area. The capacity of
both 6” pipes is 2 * 36 * p (A = pr2) or 72 p.
The cross sectional area of the single pipe
must be the same or 72 p must be equal to pr2 where r is the new radius. Solving that problem will yield an r of 8.49 inches.
Problem #8
8. Each side of the small squares
below is a toothpick. Cross out six toothpicks, so that there remain only two
squares.
One possible solution is pictured below. The red toothpicks have been removed.
Problem #9
9. Determine the largest number of
boxes of dimension 2x2x3 that can be placed inside a
Although the volumes have a ratio of 5 to 1, by trial and error you
will find that only 4 of the smaller box can fit into the larger box.
Problem #10
10. In triangle ABC: AB = AC, angle A = 80°, CE = CD and BF = BD then angle EDF
equals what measure?
