Earth’s Oceans Part II Source: CK12.org Earth Science Chapter 14 Author: Robert G. Smith.
CK12.org Algebra 1 Chap 1 Equations and Functions
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Transcript of CK12.org Algebra 1 Chap 1 Equations and Functions
+, ,×,÷x, y, a, b, c, . . .
x x
x
a + b = b + a a
b
x 3x + 1 = 10
x
3x− 10 f(x) = 3x− 10 f x
4P
P = l + w + l + w
2 2
P = 2× + 2×
· ×
P = 2 · l + 2 · w
11x
11 · x 11× x P
P = 2 + 2
A = × → A = · → A =
2 +2 P = 2 +2 .
(=)
x = 12 2x− 712 x x
12
2x− 7 = 2(12)− 7
= 24− 7
= 17
2 · 12 = 2(12) 6= 212
x = −1 −9x + 2
−9(−1) + 2 = 9 + 2
= 11
y = −2 7y− 11 + 2
7
(−2)− 11(−2) + 2 = −3
1
2+ 22 + 2
= 24− 31
2
= 201
2
( ) ( )
A = h2 (a + b)
a = 10 , b = 15 h = 8
a, b h
A
A =h
2(a + b) 10 , 15 8 .
A =8
2(10 + 15) . (10 + 15) = 25;
8
2= 4
A = 4(25) = 100
100
19 (5x + 3y + z) x = 7 y = −2 z = 11
x y z
1
9(5(7) + 3(−2) + (11)) .
1
9(35 + (−6) + 11) .
1
9(40) =
40
9≈ 4.44
≈ 4.44
R1R2
R1 + R2
R1 R2
30 15 .
R1R2
R1 + R2R1 = 30 R2 = 15.
(30)(15)
30 + 15=
450
45= 10
10 .
2 · 2 = 22
2 · 2 · 2 = 23
22 = 4
23 = 8
x8
x · x · x · x · x · x · x · x
A = πr2 r = 17
A = πr2 17 .
A = π(17)2 π · 17 · 17 = 907.9202 . . . 2 .
907.92
5x2 − 4y x = −4 y = 5
5x2 − 4y = 5(−4)2 − 4(5) x = −4 y = 5.
= 5(16)− 4(5) (−4)2 = 16.
= 80− 20
= 60
2x2 − 3x2 + 5 x = −5x
2x2 − 3x2 + 5 = 2(−5)3 − 3(−5)2 + 5 − 5 .
= 2(−125)− 3(25) + 5 (−5)3 = (−5)(−5)(−5) = −125 (−5)2 = (−5)(−5) = 25
= −250− 75 + 5
= −320
x2y3
x3+y2 x = 2 y = −4x y
x2y3
x3 + y2=
(2)2(−4)3
(2)3 + (−4)22 − 4 .
4(−64)
8 + 16=−256
24=−32
3: (2)2 = (2)(2) = 4 (2)3 = (2)(2)(2) = 8.
(−4)2 = (−4)(−4) = 16 (−4)3 = (−4)(−4)(−4) = −64.
(h) h = −32t2 + 60t + 20(t) t = 2
h = −32t2 + 60t + 20 2 .
= −32(2)2 + 60(2) + 20
= −32(4) + 60(2) + 20
= 12
2× 11x
1.35 · y
3× 14
14 · z
a = −3, b = 2, c = 5 d = −4
2a + 3b
4c + d
5ac− 2b
2ac−d
3bd
a−4b3c+2d
1a+b
abcd
x = −1, y = 2, z = −3, w = 4
8x3
5x2
6z3
3z2 − 5w2
x2 − y2
z3+w3
z3−w3
2x2 − 3x2 + 5x− 4
4w3 + 3w2 − w + 2
3 + 1z2
C x C = 2000 + 3x
1000
2000
V = 4x(10 − x)2 x
V
x = 2
x = 3
22x
1.35y
34
z4
0
16
−79
−23
−32
−117
−1
310
−8
−5162
−53
−3
37−91
−14
302
319
$5000
$8000
512 in3
588 in3
2 + 4× 7− 1 =?
2 + 4 = 6
6× 7 = 42
42− 1 = 41
41
29
2 + (4× 7)− 1 =?
4× 7 = 28
2 + (28)− 1 =?
2 + 28 = 30
30− 1 = 29
32 x3
4− 7− 11− 24− (7− 11) + 24− [7− (11 + 2)]
4− 7− 11 + 2 = −3− 11 + 2
= −14 + 2
= −12
7− 11 = −4
4− (7− 11) + 2 = 4− (−4) + 2
= 8 + 2
= 10
(11 + 2) = 13
4− [7− (11 + 2)] = 4− [7− (13)]
= 4− [−6]
= 10
3× 5− 7÷ 23× (5− 7)÷ 2(3× 5)− (7÷ 2)
3× 5 = 15 7÷ 2 = 3.5
3× 5− 7÷ 2 = 15− 3.5
= 11.5
5− 7 = −2
3× (5− 7)÷ 2 = 3× (−2)÷ 2
= (−6)÷ 2
= −3
3× 5 = 15 7÷ 2 = 3.5
(3× 5)− (7÷ 2) = 15− 3.5
= 11.5
8− [19− (2 + 5)− 7)]
8− (19− (2 + 5)− 7) = 8− (19− 7− 7)
= 8− 5
= 3
2− (3x + 2) x = 23y2 + 2y − 1 y = −32− (t− 7)2 × (u3 − v) t = 19, u = 4 v = 2
x
2− (3(2) + 2) 3(2) 3× 2
2− (3× 2 + 2) = 2− (6 + 2) .
2− 8 = −6 :
y
3× (−3)2 + 2× (−3)− 1
= 3× (−3)2 + 2× (−3)− 1 : (−3)2 = 9
= 3× 9 + 2× (−3)− 1 : 3× 9 = 27; 2×−3 = −6
= 27 + (−6)− 1 .
= 27− 6− 1
= 20
t u v
2− (19− 7)2 × (43 − 2)
= 2− (19− 7)2 × (43 − 2) : (19− 7) = 12; (43 − 2) = (64− 2) = 62
= 2− 122 × 62 : 122 = 144
= 2− 144× 62 : 144× 62 = 8928
= 2− 8928 .
= −8926
z+34 − 1 z = 2(
a+2b+4 − 1
)
+ b a = 3 b = 1
2×(
w+(x−2z)
(y+2)2− 1)
w = 11 x = 3 y = 1 z = −2
z
2 + 3
4− 1
(2 + 3)
4− 1
5
4− 1
54
5
4= 1
1
4
5
4− 1 = 1
1
4− 1 =
1
4
a b
(
3 + 2
1 + 4− 1
)
− 1
(3 + 2) (1 + 4)
(
5
5− 1
)
− 1 .
(1− 1)− 1 .
0− 1 = −1
w x y z
2
(
[11 + { 3− 2(−2) }][{ 1 + 2 }2]
− 1
)
{} .
{ 1 + 2 } = 3; { 3− 2(−2) } = 3 + 4 = 7(
[11 + 7]
[32]− 1
)
.
2
(
18
9− 1
)
.
2(2− 1) .
2(1) = 2
[3(x2 − 1)2x4 + 12] + 5x3 − 1 x = −3
x = −3
[3((−3)2 − 1)2(−3)4 + 12] + 5(−3)3 − 1
−13
[3(x2 − 1)2x4 + 12] + 5x3 − 1 x = −3
x = −3 −3 x x
x
−13
x = 2, 23
x = 2 x 23 x
62
x = 23 x 2
3 x
13.21 107081
3x2−4y2+x4
(x+y)2x = −2 y = 1
x y −2 x 1 y x y
÷−.88̄ −8
9
8− (19− (2 + 5)− 7)
2 + 7× 11− 12÷ 3
(3 + 7)÷ (7− 12)
2·(3+(2−1))4−(6+2) − (3− 5)
jk
j+kj = 6 k = 12
2y2 x = 1 y = 5
3x2 + 2x + 1 x = 5
(y2 − x)2 x = 2 y = 1
4x9x2
−3x+1 x = 2
z2
x+y+ x2
x−yx = 1 y = −2 z = 4
4xyz
y2−x2 x = 3 y = 2 z = 5
x2−z2
xz−2x(z−x) x = −1 z = 3
5− 2 · 6− 4 + 2 = 5
12÷ 4 + 10− 3 · 3 + 7 = 11
22− 32− 5 · 3− 6 = 30
12− 8− 4 · 5 = −8
x2 + 2x− xy x = 250 y = −120
(xy − y4)2 x = 0.02 y = −0.025x+y−z
xy+yz+xzx = 1
2 y = 32 z = −1
(x+y)2
4x2−y2 x = 3 y = −5d
3
75
−2
−2
4
300
86
3
831
− 473
−24
− 85
(5− 2) · (6− 5) + 2 = 5
(12÷ 4) + 10− (3 · 3) + 7 = 11
(22− 32− 5) · (3− 6) = 30
12− (8− 4) · 5 = −8
93000
0.00000025
− 125
411
$12
1 2 3 4 5 6 7
($) 12 24 36 48 60 72 84
x x = 2.5
= 12×
x =y =
y = 12x
x y
x y
$12 1296
( ) = 12× ( )
( ) = 1296
y = ( )
y = 12× 1296 .
y = 15552 $15552
$15552 $12
1555212 = 1296
b a 750
a : 0 10 20 30 40 50
b : 20 40 60 80 100 120
a 10b 20 b a a = 0 b = 20
a b′
a a 20.
b → b =a → 2a
20 → +20
b = 2a + 20 .
750 → b = 2(750) + 20
b = 2(750) + 20
b = 1500 + 20 = 1520
b a 1520 = 2(750) + 20
x− y −−2 10
0 0
2 −10
4 −20
6 −30
y x y x
x x y
y x
y → y
x → −5x
y = −5x .
x y
x
−2 0 2 4 6 y
x = −2 : y = −5(−2) y = +10
x = 0 : y = −5(0) y = 0
x = 2 : y = −5(2) y = −10
x = 4 : y = −5(4) y = −20
x = 6 : y = −5(6) y = −30
x y
$100
1 1002 783 564 34
$22
1 22
( ) = 1, ( ) = 100
= n
= b
n 1 b
22→ b = −22n+?
−22 −44 −66, . . .
1 100
100 = −22(1)+?
100 22 122
b = −22n + 122 .
n
n = 1 → b = −22(1) + 122 → b = 122− 22 = 100
n = 2 → b = −22(2) + 122 → b = 122− 44 = 78
n = 3 → b = −22(3) + 122 → b = 122− 66 = 56
n = 4 → b = −22(4) + 122 → b = 122− 88 = 34
25 23
2523
x =25
→ x
25 → x− 2523 → 2
3x
x− 25 =2
3x .
25
x− 25 =2
3x
x− 25 + 25 =2
3x + 25
x =2
3+ 25
23x
x− 2
3x =
2
3x− 2
3x + 25
1
3x = 25
3
3 · 1
3x = 25 · 3
x = 75
x
75
75 25 75− 25 = 50
23
23 · 75 = 50
23
1 202 403 604 805 100
10
24
9
35
25 2350
35
25
40 20
3 20$120
P = 20t P = t = P = t =
= 200
y = 24− x y = x =
15
x = 7x = 35 = 5
x = 2x + 25 = 5x + 5(2x + 25) = 350 = 15 55
x = x + 1 = x + x + 1 = 35 = 17 = 18
x = x = 3(x− 6) 9
3
$150
y = 12x
y ≤ 12x
3x− 2 = 5x + 9 = 2x + 5x3 = 15x2 + 1 = 10
>
≥<
≤6=
3x < 54− x ≤ 2x
x2 + 2x− 1 > 03x4 ≥ x
2 − 3
12 209 33
21$20
n =
12 20
n + 12 = 20
n + 12 = 20
n =
9 339 33
2× n− 9 = 33
2n− 9 = 33
n =
2121
4× n + 5 = 21
4n + 5 = 21
m =
$20
20 = 14 ×
14 = 20
2420 $12
t =
2424
2 = 24× t
t = 224 t = 1
12 t = 112 × 60 = 5
5
p =
20 $12−20 $12
p− 0.20p = 12
0.8p = 12 p = 120.8 o = 15
$15
20 0.20× $15 = $3$18− $3 = $15
5 2150
8242
n =
5 + n ≤ 2
d =
d < 150
x =
x > 82
h =
h ≥
x = 5 3x + 2 = −2x + 27
x = 5 5 x
3x + 2 = −2x + 27
3 · x + 2 = −2 · x + 27
3 · 5 + 2 = −2 · 5 + 27
15 + 2 = −10 + 27
17 = 17
x = 5 3x + 2 = −2x + 27
y = −1 3y + 5 = −2y
z = 3 z2 + 2z = 8
x = −12 3x + 1 = x
3(−1) + 5 = −2(−1)
−3 + 5 = 2
2 = 2
y = −1 3y + 5 = −2y
32 + 2(3) = 8
9 + 6 = 8
15 = 8
z = 3 z2 + 2z = 8
3
(
−1
2
)
+ 1 = −1
2(
−3
2
)
+ 1 = −1
2
−1
2= −1
2
x = 12 3x + 1 = x
a = 10 20a ≤ 250b = −2 3−b
b> −4
x = 34 4x + 5 ≤ 8
z = 25 z5 + 1 < z − 20
20(10) ≤ 250200 ≤ 250
a = 10 20a ≤ 250a = 10 20
a ≤ 12.5.
12.5
3− (−2)
(−2)>− 4
3 + 2
−2>− 4
−5
2>− 4
−2.5 >− 4
b = −2 3−bb
> −4
4
(
3
4
)
+ 5 ≥ 8
3 + 5 ≥ 8
8 ≥ 8
x = 34 4x + 5 ≥ 8
25
5+ 1 < 25− 20
5 + 1 < 5
6 < 5
z = 25 z5 + 1 < z − 20
$0.50 $2.00$8
a =
a + 6 =
$0.50 $2.00 $8$0.50 $2 $8
0.5× (a + 6) + 2× a = 8
0.5a + 0.5× 6 + 2a = 8 .
2.5a + 3 = 8
2.5a = 5
a = 2
a =a + 6 = 2 + 6 = 8
2 8
2× $2 + 8× $0.50 = $4 + $4 = $8
24
x =
24
2× 24 ≤ x
48 ≤ x
48x = 48 48 48
$10 $0.20 $25
$200 $4$324
$55 $0.45 $100
4 7
65
54
5$250
$0.49 $3
a = −3 4a + 3 = −9
x = 43
34x + 1
2 = 32
y = 2 2.5y − 10.0 = −5.0
z = −5 2(5− 2z) = 20− 2(z − 1)
x = 12 2(x + 6) ≤ 8x
z = −9 1.4z + 5.2 > 0.4z
y = 40 −52y + 1
2 < −18
t = 0.4 80 ≥ 10(3t + 2)
27 $15000
$10006 $1200 5 $2000
$2000
x = 25 = 10 + 0.2x
p = 324 = 200 + 4p
m = 55 + 0.45 = 100
n = n = 4 + 7
x = x ≤ 65
n = 2 + 1 < 54
P = 0.05P ≥ 250
n = 0.49 ≤ 3
4(−3) + 3 = −9 −12 + 3 = −9 −9 = −934
(
43
)
+ 12 = 3
2 1 + 12 = 3
232 = 3
2
2.5(2)− 10.0 = −5.0 5.0− 10.0 = −5.0 −5.0 = −5.0
2(5− 2(−5)) = 20− 2((−5)− 1) 2(5 + 10) = 20− 2(−6) 2(15) = 20 + 12 30 = 32
2(12 + 6) ≤ 8(12) 2(18) ≤ 96 36 ≤ 96
1.4(−9) + 5.2 > 0.4(−9) −12.6 + 5.2 > −3.6 −7.4 > −3.6
− 52 (40) < −18 −100 + 1
2 < −18 −99.5 < −18
80 ≥ 10(3(0.4) + 2) 80 ≥ 10(1.2 + 2) 80 ≥ 10(3.2) 80 ≥ 32
x = 0.27x = 15000 x = $55556
x = 1000 + 0.06x > 1200 + 0.05(x− 2000) x > 10000.
$2
$8 $2
2 8
x =y =
x y f()2
8
f(x) = 2x + 8
f(x) = 2x + 8 y f(x)y f(x)
$8 $2
f(x) = 2x + 8 = y
x y
x
2.25
y
(0, 1, 2, 3, 4, 5, 6, )x
f(x) = 2x + 8 = y
y = 8, 10, 12, 14, 16, 18, 20, . . .
8
75y = x2
x =y =
y = f(x) = 0.75x
x
y
x
y = x2 y x
x
y
x
y = f(x) = 3x
R = f(w) = 3w
v = f(t) = 3t
t d v y
x
$8 $2
2× +80, 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 18, 20
x y
0 8
1 10
2 12
3 14
4 16
5 18
6 20
f(x) = 5x− 9 −4,−3,−2,−1, 0, 1, 2, 3, 4
f(x) = 1x
−1,−0.5,−0.2,−0.1,−0.01, 0.01, 0.1, 0.2, 0.5, 1
x f(x) = 5x− 9 = y
4 5(−4)− 9 = −29
− 3 5(−3)− 9 = −24
− 2 5(−2)− 9 = −19
− 1 5(−1)− 9 = −14
0 5(0)− 9 = −− 9
1 5(1)− 9 = −4
2 5(2)− 9 = 1
3 5(3)− 9 = 6
4 5(4)− 9 = 11
x f(x) =1
x= y
− 11
−1= −1
− 0.51
−0.5= −2
− 0.21
−0.2= −5
− 0.11
−0.1= −10
− 0.011
−0.01= −100
0.011
0.01= 100
0.11
0.1= 10
0.21
0.2= 5
0.51
0.5= 2
1.01
1= 1
2 4 6 8 10
($) 24 48 72 86 120
$24 2 $48 4 $120 10 $12
= $12× f(x) = 12x
x −3 −2 −1 0 1 2 3
y 3 2 1 0 1 2 3
f(x) = |x|
$11.95$0.50
x =y =
$11.95 $0.50= + ×
y = f(x) = 11.95 + 0.50x
12 feet × 24 feet
$3
x =y =
= 12x + 12x + 24x + 24x + x2 + x2 + x2 + x2 + 72x + 4x2
f(x) = 3(72x + 4x2) = 216x + 12x2
x =y =
1 2 3 4 5 6 7
2 4 8 16 32 64 128
1 : 2
2 : 2× 2 = 4
3 : 2× 2× 2 = 8
4 : 2× 2× 2× 2 = 16
2 = 21
2× 2 = 22
2× 2× 2 = 23
y = f(x) = 2x
$10
$25 $15
f(x) = 15x− 12
f(x)2x2 + 5
f(x) = 1x
y = x2 − 5 −2,−1, 0, 1, 2
y = 2x− 34 −2.5, 1.5, 5
$6.505, 10, 15, 20, 25, 30
A = 12bh 8
1, 2, 3, 4, 5, 6
f(x) =√
2x + 3 −1, 0, 1, 2, 3, 4, 5
x 3 4 5 6
y 9 16 25 36
0 1 2 3
15 20 25 30
x 0 1 2 3
y 24 12 6 3
x
$40 $253
$2500 $12.50
15
5
0 0
−1,−4,−5
−2, 0, 74
5 10 15 20 25 30
$32.50 $65 $97.50 $130 $162.50 $195
1 2 3 4 5 6
4 8 12 16 20 24
x − 1 0 1 2 3 4 5
y 1 1.73 2.24 2.65 3 3.32 3.6
y = x2
y = 15 + 5x
y = 242x
f(x) = x− 1
y = 40 + 25x $115
200
(0, 0)
x−y−
x y (x, y)(4, 2)
42
(5, 3)(−2, 6)
(3,−4)(−5,−7)
x
x
y
y
x− y−
x − 2 − 1 0 1 2
y 6 8 10 12 14
(−2, 6), (−1, 8), (0, 10), (1, 12), (2, 14)
0 1 2 3 4
0 1 4 9 16
(0, 0), (1, 1), (2, 4), (3, 9), (4, 16).
x = 0
12 13 14 15 16
41 44 47 50 53
$3$5
(12, 41), (13, 44), (14, 47), (15, 50), (16, 53).x−
12 12
f(x) = |x− 2|
x y = f(x) = |x− 2|− 4 | − 4− 2| = | − 6| = 6
− 3 | − 3− 2| = | − 5| = 5
− 2 | − 2− 2| = | − 4| = 4
− 1 | − 1− 2| = | − 3| = 3
0 |0− 2| = | − 2| = 2
1 |1− 2| = | − 1| = 1
2 |2− 2| = |0| = 0
3 |3− 2| = |1| = 1
4 |4− 2| = |2| = 2
5 |5− 2| = |3| = 3
6 |6− 2| = |4| = 4
7 |7− 2| = |5| = 5
8 |8− 2| = |6| = 6
f(x) =√
x
x y = f(x) =√
x
0√
0 = 0
1√
1 = 1
2√
2 ≈ 1.41
3√
3 ≈ 1.73
4√
4 = 2
5√
5 ≈ 2.24
6√
6 ≈ 2.45
7√
7 ≈ 2.65
8√
8 ≈ 2.83
9√
9 = 3
41 173.5
x y
0 0
0.2 41
0.5 41
0.8 41
1 41
1.2 58
1.5 58
1.8 58
2 58
2.2 75
2.5 75
2.8 75
3.0 75
3.2 92
3.5 92
0 1 2 3 4 5 6
0 1.5 3 4.5 6 7.5 9
1.5
= 1.5×
f(x) = 1.5x
x − 4 − 3 − 2 − 1 0 1 2 3 4
y 8 4.5 2 .5 0 .5 2 4.5 8
y
x − 4 − 3 − 2 − 1 0 1 2 3 4
y16
2
9
2
4
2
1
2
0
2
1
2
4
2
9
2
16
2
y x 2y = 1
2x2 f(x) = 12x2
− 1 0 1 2 3 4 5
10 5 2.5 1.2 0.6 0.3 0.15
0 : = 5
1 : = 5 · 1
2
2 : = 5 · 1
2· 1
2
3 : = 5 · 1
2· 1
2· 1
2
f(x) = 5(
12
)x
1900 285
1930 300
1950 310
1990 350
f(x) = 0.0066x2 − 24.9x + 23765
x− y−x y
(1, 3), (−1,−2), (3, 5), (2, 5), (3, 4)(−3, 20), (−5, 25), (−1, 5), (7, 12), (9, 2)
x 2 4 6 8 10
y 41 44 47 50 53
x 2 1 0 1 2
y 12 10 8 6 4
x−x y−
(1, 3), (−1,−2), (3, 5), (2, 5), (3, 4)y− x− 3
(−3, 20), (−5, 25), (−1, 5), (7, 12), (9, 2)x y−
x 2 4 6 8 10
y 4 4 4 4 4
x
x 2 1 0 1 2
y 12 10 8 6 4
y− x− 2 y−x− 1
(4,−4)
(2, 7)
(−3,−5)
(6, 3)
(−4, 3)
x − 10 − 5 0 5 10
y − 3 − 0.5 2 4.5 7
0 1 2 3
( 3) 0 1 8 27
− 2 − 1 0 1 2
50 25 0 25 50
$50 $8
f(x) = (x− 2)2
f(x) = 3.2x
(1, 7), (2, 7), (3, 8), (4, 8), (5, 9)
(1, 1), (1,−1), (4, 2), (4,−2), (9, 3), (9,−3)
x − 4 − 3 − 2 − 1 0
y 16 9 4 1 0
20 25 25 30 35
3 4 7 4 2
1991
1996
2004
2005
1940
1955
1980
1995
10
17
10
17
(−6, 4)
(7, 6)
(−8,−2)
(4,−7)
(5, 0)
f(x) = 12 |x|
f(x) =√
x
27.5
35.6
22.2
23
63
69
74
76
$19, 500
$56, 000
$10, 000
$35, 000
5040 $21
50 40$21
50 0.5×= 0.5×
40 60 0.6× = 0.6 × (0.5×) = 0.3×
= 0.3×$21
0.3× = $21
0.3× = $21
$21 0.3
= $21÷ 0.3 = $70
$70
$70
50 $70 = .5× $70 = $35
= $70− $35 = $35
40 $35 = .4× $35 = $14
$35− $14 = $21
$35 7.75
5 $45, 000
$250 14 × 189 × 10
15 $10$65
$250 $2025
$12 $1.50 $24
$2.95 $3.50 $4.2522 26 15
180
$37.71
$42857
$89.29
$45.25
$750
8
$219.65
30◦ 60◦ 90◦
$1
$1
$1
1012 2
1012
2 6
10 12
1 10 602 12 72
2 6
1 10 602 12 723 14 844 16 965 18 1086 20 120
120
12
60 12
= 60 + 12× 5 = 120
8
8
= 2 + 1 = 3
= 3 + 3 = 6
1 12 33 6
3 3
1 12 33 64 6 + 4 = 105 10 + 5 = 156 15 + 6 = 217 21 + 7 = 288 28 + 8 = 36
36 8
81 1 2 2 3 3 4 4 5 5 6
6 7 7 8 81 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36
$180 $10 $20 12
$18012 $10 $20
$10 $20$10 $20 $10 $20
$10 $20
12 0 $10(12) + $20(0) = $12011 1 $10(11) + $20(1) = $13010 2 $10(10) + $20(2) = $1409 3 $10(9) + $20(3) = $1508 4 $10(8) + $20(4) = $1607 5 $10(7) + $20(5) = $1706 6 $10(6) + $20(6) = $1805 7 $10(5) + $20(7) = $1904 8 $10(4) + $20(8) = $2003 9 $10(3) + $20(9) = $2102 10 $10(2) + $20(10) = $2201 11 $10(1) + $20(11) = $2300 12 $10(0) + $20(12) = $240
$10 $20$10 $20
$10 $20
$10 $20 = 6($10) + 6($20) = $60 + $120 = $180
$18012 $10 $20
$10 $20$10 $20 $10 $20
$10 $20
12 0 $10(12) + $20(0) = $12011 1 $10(11) + $20(1) = $130
$10 $20
10 2 $10(10) + $20(2) = $140
$10 $20$10 $140
$40 $10$20
$10 $20
6($10) + 6($20) = $180
$10 $20
$10 $20 6($10) + 6($20) = $60 + $120 = $180.
20 × 20
20 × 20
×× ×
1
118 18
× ×18× 18× 1 = 324 in3
1 1 18 18 18× 18× 1 = 3242 2 16 16 16× 16× 2 = 5123 3 14 14 14× 14× 3 = 5884 4 12 12 12× 12× 4 = 5765 5 10 10 10× 10× 5 = 5006 6 8 8 8× 8× 6 = 3847 7 6 6 6× 6× 7 = 2528 8 4 4 4× 4× 8 = 1289 9 2 2 2× 2× 9 = 3610 10 0 0 0× 0× 10 = 0
10
588 in3
588 in3
2.5 2.5 15 15 15 × 15 × 2.5 =562.5
2.6 2.6 14.8 14.8 14.8× 14.8× 2.6 =569.5
2.7 2.7 14.6 14.6 14.6× 14.6× 2.7 =575.5
2.8 2.8 14.4 14.4 14.4× 14.4× 2.8 =580.6
2.9 2.9 14.2 14.2 14.2× 14.2× 2.9 =584.8
3 3 14 14 14× 14× 3 = 5883.1 3.1 13.8 13.8 13.8× 13.8× 3.1 =
590.43.2 3.2 13.6 13.6 13.6× 13.6× 3.2 =
591.93.3 3.3 13.4 13.4 13.4× 13.4× 3.3 =
592.53.4 3.4 13.2 13.2 13.2× 13.2× 3.4 =
592.43.5 3.5 13 13 13 × 13 × 3.5 =
591.5
3.3
3.3
3.3
$2.25 40
12th
624 21 18
5 10
75 55
12 15
24
$30
5 35
23
7
50
5.5
5
3 in3
