Homework Assignment #1
0.1.1.a) 1/4 0.1.1.b) 3/16
0.1.1.c) 5/16 0.1.1.d) 1-7/16
0.1.2.a) Remove
the middle ninth of the square.
0.1.2.b) 
0.1.2.c) 0, 1/9, 17/81, 217/729
0.1.2.d) 1, 8/9, 64/81, 512/729
0.1.3.a) 8 ÷ 4 = 2
0.1.3.b) 8 × 1/4 = 2
0.1.3.c) 3/4 × 8 = 6
0.1.4.a) 9/16
0.1.4.b) 9/16 × 12 = 27/4 = 6.75
0.1.5.a) 1/9
0.1.5.b) 1/81
0.1.5.c) 4/9 + 3/81 =
0.1.6.a) 1/4 × (1/4 × (32)) = 2
0.1.6.b) 3/4 × (1/4 × (1/4 × (32))) = 3/2
0.1.6.c) 1/2 × (1/2 × (1/4 × (32))) = 2
PROBLEM
SET 0.2
0.2.1.a) 4
0.2.1.b) 8 = 23
0.2.1.c) 32 = 25
0.2.1.d) 2n
0.2.2.a) 5 × 1/4 = 5/4 = 1.25
0.2.2.b) 5 × 1/8 = 5/8 = 0.625
0.2.2.c) 5× 1/16 = 5/16 = 0.3125
0.2.2.d) 5 × (1/2)2
0.2.2.e) 5 × (1/2)4
0.2.2.f) 5 × (1/2)n
0.2.3.a) 64
0.2.3.b) 512
0.2.3.c) 82, 83
0.2.3.d) 86
0.2.3.e) n
– 1
0.2.3.f) No, this would give a negative
exponent and thus a fractional area.
0.2.4.a) 10
0.2.4.b) 50
0.2.4.c) 2 × 52
0.2.4.d) On each branch you add two branches. These two branches break the tree into 5 sub branches.
0.2.5.a
& b)
|
Stage |
Area of one shaded
triangle |
Total area of shaded
triangles |
|
0 |
1 |
1 |
|
1 |
1/4 |
3/4 |
|
2 |
1/16 |
9/16 |
|
3 |
1/64 |
27/64 |
0.2.5.c) The area of one triangle is 1/4n
for the n-th stage. The total area of the shaded triangles is 1 – 1/4n
or 3n/4n
PROBLEM
SET 0.3
0.3.1.a)
|
|
Total length of each stage |
|||
|
Stage |
Without exponent |
With exponent |
Decimal number |
|
|
0 |
1 |
10 |
1 |
|
|
1 |
5 × 1/4 = 5/4 |
51 × (1/4)1=5/4 |
1.25 |
|
|
2 |
25 × 1/16 = 25/16 |
52 × (1/4)2 |
1.56 |
|
|
3 |
125 × 1/64 = 125/64 |
53 × (1/4)3 |
1.95 |
|
0.3.1.b) Stage 5, (5/4)5 =
3125/1024 > 3
0.3.1.c) Stage 11
0.3.2.a)
|
|
Total length of each stage |
||
|
Stage |
Without exponent |
With exponent |
Decimal number |
|
0 |
1 |
10 |
1 |
|
1 |
6 × 1/4 = 6/4 = 3/2 |
61 × (1/4)1
= (3/2)1 |
1.50 |
|
2 |
36 × 1/16 = 36/16 = 9/4 |
62 × (1/4)2
= (3/2)2 |
2.25 |
|
3 |
216 × 1/64 = 216/64 = 27/8 |
63 × (1/4)3
= (3/2)3 |
3.375 |
0.3.2.b) Stage 6
0.3.2.c) Stage 12
0.3.3.a)
|
|
Total length of each stage |
||
|
Stage |
Without exponent |
With exponent |
Decimal number |
|
0 |
1 |
10 |
1 |
|
1 |
7 × 1/3 = 7/3 |
71 × (1/3)1
= (7/3)1 |
2.33 |
|
2 |
49 × 1/9 = 49/9 |
72 × (1/3)2
= (7/3)2 |
5.44 |
|
3 |
343 × 1/27 = 343/27 |
73 × (1/3)3
= (7/3)3 |
12.70 |
|
4 |
2401 × 1/81 = 2401/81 |
74 × (1/3)4
= (7/3)4 |
29.64 |
0.3.3.b) Stage 5
0.3.3.c) No, 168 is not a power of 7/3.
0.3.4.a)
|
|
Total length of each stage |
||
|
Stage |
Without exponent |
With exponent |
Decimal number |
|
0 |
1 |
10 |
1 |
|
1 |
8 × 1/3 = 8/3 |
81 × (1/3)1
= (8/3)1 |
2.67 |
|
2 |
64 × 1/9 = 64/9 |
82 × (1/3)2
= (8/3)2 |
7.11 |
|
3 |
512 × 1/27 = 512/27 |
83 × (1/3)3
= (8/3)3 |
18.96 |