POLINOMIAL pangkat 3

Mata Kuliah Rekyasa Perhitungan Cadangan

PERHITUNGAN CADANGAN DENGAN

METODE POLINOMIAL PANGKAT TIGA

OLEH :

MUHAMMAD SUKARDI

D62107026

PROGRAM STUDI TEKNIK PERTAMBANGAN FAKULTAS TEKNIK

UNIVERSITAS HASANUDDIN

MAKASSAR 2010

A. PENAMPANG 1-1’

1. Nilai Segmen

PENAMPANG 1-1’ SEGMEN X Y

0 0 70 1 25 75 2 50 50 3 75 65 4 100 55 5 125 60 6 150 35

2. Diketahui

H0 = X1 - X0 = 25.00 H1 = X2 - X1 = 25.00 H2 = X3 - X2 = 25.00 H3 = X4 - X3 = 25.00 H4 = X5 - X4 = 25.00 H5 = X6 - X5 = 25.00

3. Maka matriks [H] [C] = [A] adalah :

4. Perhitungan Eliminasi Gauss Jordan membuat matriks baru

M21 = H21/H11 = 25.00

a21 baru = a21 - M21 (a11) = 25 - (25 x 1) = 0.00 a22 baru = a22 - M21 (a12) = 25 - (25 x 0) = 100.00 a23 baru = a23 - M21 (a13) = 25 - (25 x 0) = 25.00 A21 baru = A21 - M21 (A11) = 3.6 - (25 x 0) = -3.60

M32 = H32/H22 = 0.25

a0 = 70.00 a1 = 75.00 a2 = 50.00 a3 = 65.00 a4 = 55.00 a5 = 60.00 a6 = 35.00

a32 baru = a32 - M32 (a22) = 25 - (0.25 x 100) = 0 a33 baru = a33 - M32 (a23) = 100 - (0.25 x 25) = 93.75 a34 baru = a34 - M32 (a24) = 25 - (0.25 x 25) = 25.00 A31 baru = A31 - M21 (A21) = 4.8 - (0.25 x-3.6) = 5.70

M43 = H43/H33 = 0.25

a43 baru = a43 - M43 (a33) = 25 - (0.25 x 100) = 0 a44 baru = a44 - M43 (a34) = 100 - (0.25 x 25) = 93.75 a45 baru = a45 - M43 (a35) = 25 - (0.25 x 25) = 25.00 A41 baru = A41 - M43 (A31) = 3 - (0.25 x 4.8) =

-4.20

M54 = H54/H44 = 0.25

a54 baru = a54 - M54 (a44) = 25 - (0.25 x 100) = 0 a55 baru = a55 - M54 (a45) = 100 - (0.25 x 25) = 93.75 a56 baru = a56 - M54 (a46) = 25 - (0.25 x 25) = 25.00 A51 baru = A51 - M54 (A41) = 1.8 - (0.25 x 3) =

2.55

Matriks baru

5. Substitusi mundur untuk mendapatkan nilai C

C5 = 0

93.75C4 + 25C5 = 2.55 C4 = 0.02720 93.75C3 + 25C4 = - 4.2 C3 = -0.05205

93.75C2 + 25C3 = 5.7 C2 = 0.07468 93.75C1 + 25C2 = - 3.6 C1 = -0.05467

C0 = 0

6. Mencari nilai b dengan rumus :

b0 = 0.66

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b1 = -0.71 b2 = -0.21 b3 = 0.24 b4 = -0.25 b5 = -1.00

7. Mencari nilai d dengan rumus :

d0 = -0.000729 d1 = 0.001725 d2 = -0.001690 d3 = 0.001057 d4 = -0.000363 d5 = 0.000000

8. Perhitungan luas penampang :

A0 = 1883.69 A1 = 1536.44 A2 = 1408.04 A3 = 1532.36 A4 = 1402.08 A5 = 1187.50

Luas penampang 1-1’ diperoleh dari penjumlahan A0, A1, A2, A3, A4 dan A5

A penampang 1-1’ = A0+ A1+ A2+ A3+ A4+ A5

= 8.950,11 m2

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B. PENAMPANG 2-2’

1. Nilai Segmen

PENAMPANG 2 SEGMEN X Y

0 0 60 1 25 80 2 50 55 3 75 70 4 100 55 5 125 70 6 150 45

2. Diketahui

H0 = X1 - X0 = 25.00 a0 = 60.00

H1 = X2 - X1 = 25.00 a1 = 80.00

H2 = X3 - X2 = 25.00 a2 = 55.00

H3 = X4 - X3 = 25.00 a3 = 70.00

H4 = X5 - X4 = 25.00 a4 = 55.00

H5 = X6 - X5 = 25.00 a5 = 70.00

a6 = 45.00



M21 = H21/H11 = 25.00


M32 = H32/H22 = 0.25

a32 baru = a32 - M32 (a22) = 25 - (0.25 x 100) = 0 a33 baru = a33 - M32 (a23) = 100 - (0.25 x 25) = 93.75 a34 baru = a34 - M32 (a24) = 25 - (0.25 x 25) = 25.00

A31 baru = A31 - M21 (A21) = 4.8 - (0.25 x5.4) = 6.15

M43 = H43/H33 = 0.25

a43 baru = a43 - M43 (a33) = 25 - (0.25 x 100) = 0 a44 baru = a44 - M43 (a34) = 100 - (0.25 x 25) = 93.75 a45 baru = a45 - M43 (a35) = 25 - (0.25 x 25) = 25.00 A41 baru = A41 - M43 (A31) = 3.6 - (0.25 x 4.8) = -4.80

M54 = H54/H44 = 0.25

a54 baru = a54 - M54 (a44) = 25 - (0.25 x 100) = 0 a55 baru = a55 - M54 (a45) = 100 - (0.25 x 25) = 93.75 a56 baru = a56 - M54 (a46) = 25 - (0.25 x 25) = 25.00 A51 baru = A51 - M54 (A41) = 3.6 - (0.25 x 3.6) = 4.50

Matriks baru


C5 = 0 93.75C4 + 25C5 =4.5 C4 = 0.04800

93.75C3 + 25C4 = -4.8 C3 = -

0.06400 93.75C2 + 25C3 = 6.15 C2 = 0.08267

93.75C1 + 25C2 = -5.40 C1 = -

0.07467

C0 = 0

6. Mencari nilai b dengan rumus : 11 23

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b0 = 1.4222 b1 = -0.44 b2 = -0.24 b3 = 0.07 b4 = -0.20 b5 = -1.00


d0 = -0.000996 d1 = 0.002098 d2 = -0.001956 d3 = 0.001493 d4 = -0.000640 d5 = 0.000000


A0 = 1847.22 A1 = 1677.08 A2 = 1538.19 A3 = 1583.33 A4 = 1500.00 A5 = 1437.50



= 9.583,33 m2

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C. PENAMPANG 3-3’

1. Nilai Segmen


0 0 70 1 25 75 2 50 65 3 75 85 4 100 65 5 125 80 6 150 45

2. Diketahui

a0 = 70.00 a1 = 75.00 a2 = 65.00 a3 = 85.00 a4 = 65.00 a5 = 80.00 a6 = 45.00



M21 = H21/H11 = 25.00


M32 = H32/H22 = 0.25

a32 baru = a32 - M32 (a22) = 25 - (0.25 x 100) = 0 a33 baru = a33 - M32 (a23) = 100 - (0.25 x 25) = 93.75


a34 baru = a34 - M32 (a24) = 25 - (0.25 x 25) = 25.00 A31 baru = A31 - M21 (A21) = 3,6 - (0.25 x-1,8) = 4.05

M43 = H43/H33 = 0.25


A41 baru = A41 - M43 (A31) = 4,8- (0.25 x3,6) = -5.70

M54 = H54/H44 = 0.25

a54 baru = a54 - M54 (a44) = 25 - (0.25 x 100) = 0 a55 baru = a55 - M54 (a45) = 100 - (0.25 x 25) = 93.75 a56 baru = a56 - M54 (a46) = 25 - (0.25 x 25) = 25.00 A51 baru = A51 - M54 (A41) = 4,2 - (0.25 x -4,8) = 5.40

Matriks baru


C5 = 0 93.75C4 + 25C5 =5,4 C4 = 0.05760

93.75C3 + 25C4 = -5,7 C3 = -0.07616 93.75C2 + 25C3 = 4,05 C2 = 0.06351

93.75C1 + 25C2 = -1,80 C1 = -0.03388 C0 = 0


b0 = 0.4823 b1 = -0.36 b2 = 0.38 b3 = -0.01 b4 = -0.36

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b5 = -1.40


d0 = -0.000452 d1 = 0.001298 d2 = -0.001862 d3 = 0.001783 d4 = -0.000768 d5 = 0.000000


A0 = 1856.61 A1 = 1711.42 A2 = 1891.47 A3 = 1899.17 A4 = 1737.50 A5 = 1562.50

Luas penampang 3-3 diperoleh dari penjumlahan A0, A1, A2, A3, A4 dan A5


= 10.658,67 m2

D. PENAMPANG 4-4’

1. Nilai Segmen


0 0 75 1 25 70 2 50 65 3 75 80 4 100 65 5 125 90 6 150 50

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2. Diketahui

a0 = 75.00 a1 = 70.00 a2 = 65.00 a3 = 80.00 a4 = 65.00 a5 = 90.00 a6 = 50.00



M21 = H21/H11 = 25.00

a21 baru = a21 - M21 (a11) = 25 - (25 x 1) = 0.00 a22 baru = a22 - M21 (a12) = 25 - (25 x 0) = 100.00 a23 baru = a23 - M21 (a13) = 25 - (25 x 0) = 25.00 A21 baru = A21 - M21 (A11) = 0 - (25 x 0) = 0.00

M32 = H32/H22 = 0.25

a32 baru = a32 - M32 (a22) = 25 - (0.25 x 100) = 0 a33 baru = a33 - M32 (a23) = 100 - (0.25 x 25) = 93.75 a34 baru = a34 - M32 (a24) = 25 - (0.25 x 25) = 25.00 A31 baru = A31 - M21 (A21) = 2,4 - (0.25 x 0) = 2.40

M43 = H43/H33 = 0.25


A41 baru = A41 - M43 (A31) = 3.6 - (0.25 x2,4) = -4.20

M54 = H54/H44 = 0.25


a54 baru = a54 - M54 (a44) = 25 - (0.25 x 100) = 0 a55 baru = a55 - M54 (a45) = 100 - (0.25 x 25) = 93.75 a56 baru = a56 - M54 (a46) = 25 - (0.25 x 25) = 25.00 A51 baru = A51 - M54 (A41) = 4,8- (0.25 x -3.6) = 5.70

Matriks baru


C5 = 0

93.75C4 + 25C5 = 5,70 C4 = 0.06080 93.75C3 + 25C4 = -4.20 C3 = -0.06101 93.75C2 + 25C3 = 2,40 C2 = 0.04187

93.75C1 + 25C2 = 0 C1 = -0.01047

C0 = 0


b0 = -0.1128 b1 = -0.37 b2 = 0.41 b3 = -0.09 b4 = -0.01 b5 = -1.60


d0 = -0.000140 d1 = 0.000698 d2 = -0.001372 d3 = 0.001624 d4 = -0.000811 d5 = 0.000000


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A0 = 1826.13 A1 = 1646.61 A2 = 1837.43 A3 = 1812.78 A4 = 1858.33 A5 = 1750.00



= 10.731,28 m2

E. PENAMPANG 5-5’

1. Nilai Segmen


0 0 80 1 25 75 2 50 75 3 75 75 4 100 75 5 125 100 6 150 50

2. Diketahui

a0 = 80.00 a1 = 75.00 a2 = 75.00 a3 = 75.00 a4 = 75.00 a5 = 100.00 a6 = 50.00




M21 = H21/H11 = 25.00

a21 baru = a21 - M21 (a11) = 25 - (25 x 1) = 0.00 a22 baru = a22 - M21 (a12) = 25 - (25 x 0) = 100.00 a23 baru = a23 - M21 (a13) = 25 - (25 x 0) = 25.00 A21 baru = A21 - M21 (A11) = 0.6 - (25 x 0) = 0.60

M32 = H32/H22 = 0.25

a32 baru = a32 - M32 (a22) = 25 - (0.25 x 100) = 0 a33 baru = a33 - M32 (a23) = 100 - (0.25 x 25) = 93.75 a34 baru = a34 - M32 (a24) = 25 - (0.25 x 25) = 25.00 A31 baru = A31 - M21 (A21) = 0 - (0.25 x0.6) = -0.15

M43 = H43/H33 = 0.25

a43 baru = a43 - M43 (a33) = 25 - (0.25 x 100) = 0 a44 baru = a44 - M43 (a34) = 100 - (0.25 x 25) = 93.75 a45 baru = a45 - M43 (a35) = 25 - (0.25 x 25) = 25.00 A41 baru = A41 - M43 (A31) = 0 - (0.25 x 0) = 0.00

M54 = H54/H44 = 0.25

a54 baru = a54 - M54 (a44) = 25 - (0.25 x 100) = 0 a55 baru = a55 - M54 (a45) = 100 - (0.25 x 25) = 93.75 a56 baru = a56 - M54 (a46) = 25 - (0.25 x 25) = 25.00 A51 baru = A51 - M54 (A41) = 3 - (0.25 x0) = 3.00

Matriks baru


C5 = 0

93.75C4 + 25C5 = 3 C4 = 0.03200 93.75C3 + 25C4 = 0 C3 = -0.00853

93.75C2 + 25C3 = -0.15 C2 = 0.00068

93.75C1 + 25C2 = 0,60 C1 = 0.00583 C0 = 0


b0 = -0.2486 b1 = -0.10 b2 = 0.06 b3 = -0.12 b4 = 0.47 b5 = -2.00


d0 = 0.000078 d1 = -0.000069 d2 = -0.000123 d3 = 0.000540 d4 = -0.000427 d5 = 0.000000


A0 = 1929.91 A1 = 1866.53

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A2 = 1885.23 A3 = 1844.44 A4 = 2145.83 A5 = 1875.00



= 11.546,94 m2

F. PENAMPANG 6-6’

1. Nilai Segmen


0 0 65 1 25 70 2 50 75 3 75 65 4 100 85 5 125 90 6 150 50

2. Diketahui

a0 = 65.00 a1 = 70.00 a2 = 75.00 a3 = 65.00 a4 = 85.00 a5 = 90.00 a6 = 50.00




M21 = H21/H11 = 25.00

a21 baru = a21 - M21 (a11) = 25 - (25 x 1) = 0.00 a22 baru = a22 - M21 (a12) = 25 - (25 x 0) = 100.00 a23 baru = a23 - M21 (a13) = 25 - (25 x 0) = 25.00 A21 baru = A21 - M21 (A11) = 0- (25 x 0) = 0.00

M32 = H32/H22 = 0.25

a32 baru = a32 - M32 (a22) = 25 - (0.25 x 100) = 0 a33 baru = a33 - M32 (a23) = 100 - (0.25 x 25) = 93.75 a34 baru = a34 - M32 (a24) = 25 - (0.25 x 25) = 25.00 A31 baru = A31 - M21 (A21) = 1.8 - (0.25 x0) = -1.80

M43 = H43/H33 = 0.25

a43 baru = a43 - M43 (a33) = 25 - (0.25 x 100) = 0 a44 baru = a44 - M43 (a34) = 100 - (0.25 x 25) = 93.75 a45 baru = a45 - M43 (a35) = 25 - (0.25 x 25) = 25.00 A41 baru = A41 - M43 (A31) = 3.6 - (0.25 x -1.8) = 4.05

M54 = H54/H44 = 0.25

a54 baru = a54 - M54 (a44) = 25 - (0.25 x 100) = 0 a55 baru = a55 - M54 (a45) = 100 - (0.25 x 25) = 93.75 a56 baru = a56 - M54 (a46) = 25 - (0.25 x 25) = 25.00 A51 baru = A51 - M54 (A41) = 1.8 - (0.25 x 3.6) = -2.70

Matriks baru


C5 = 0 93.75C4 + 25C5 = 0 C4 = -0.02880

93.75C3 + 25C4 = -1.8 C3 = 0.05088 93.75C2 + 25C3 = 4,05 C2 = -0.03277

93.75C1 + 25C2 = -2,70 C1 = 0.00819 C0 = 0


b0 = 0.1317 b1 = 0.34 b2 = -0.28 b3 = 0.19 b4 = 0.68 b5 = -1.60


d0 = 0.000109 d1 = -0.000546 d2 = 0.001115 d3 = -0.001062 d4 = 0.000384 d5 = 0.000000


A0 = 1676.83 A1 = 1844.50 A2 = 1726.42 A3 = 1846.25 A4 = 2225.00 A5 = 1750.00

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=11.069,00 m2

F. PENAMPANG 7-7’

1. Nilai Segmen


0 0 70 1 25 55 2 50 70 3 75 55 4 100 70 5 125 70 6 150 45

2. Diketahui

a0 = 70.00 a1 = 55.00 a2 = 70.00 a3 = 55.00 a4 = 70.00 a5 = 70.00 a6 = 45.00



M21 = H21/H11 = 25.00


a21 baru = a21 - M21 (a11) = 25 - (25 x 1) = 0.00 a22 baru = a22 - M21 (a12) = 25 - (25 x 0) = 100.00 a23 baru = a23 - M21 (a13) = 25 - (25 x 0) = 25.00 A21 baru = A21 - M21 (A11) = 3,60 - (25 x 0) = 3.60

M32 = H32/H22 = 0.25

a32 baru = a32 - M32 (a22) = 25 - (0.25 x 100) = 0 a33 baru = a33 - M32 (a23) = 100 - (0.25 x 25) = 93.75 a34 baru = a34 - M32 (a24) = 25 - (0.25 x 25) = 25.00 A31 baru = A31 - M21 (A21) = 3,6 - (0.25 x3,60) = -4.50

M43 = H43/H33 = 0.25

a43 baru = a43 - M43 (a33) = 25 - (0.25 x 100) = 0 a44 baru = a44 - M43 (a34) = 100 - (0.25 x 25) = 93.75 a45 baru = a45 - M43 (a35) = 25 - (0.25 x 25) = 25.00 A41 baru = A41 - M43 (A31) = 3.6 - (0.25 x-3,6) = 4.50

M54 = H54/H44 = 0.25

a54 baru = a54 - M54 (a44) = 25 - (0.25 x 100) = 0 a55 baru = a55 - M54 (a45) = 100 - (0.25 x 25) = 93.75 a56 baru = a56 - M54 (a46) = 25 - (0.25 x 25) = 25.00 A51 baru = A51 - M54 (A41) = 1,8 - (0.25 x 3.6) = -2.70

Matriks baru


C5 = 0 93.75C4 + 25C5 =-2,70 C4 = -0.02880

93.75C3 + 25C4 = 4,5 C3 = 0.05568 93.75C2 + 25C3 = -4,5 C2 = -0.06285 93.75C1 + 25C2 = 3,60 C1 = 0.05171

C0 = 0


b0 = -1.0309 b1 = 0.26 b2 = -0.02 b3 = -0.09 b4 = 0.48 b5 = -1.00


d0 = 0.000689 d1 = -0.001527 d2 = 0.001580 d3 = -0.001126 d4 = 0.000384 d5 = 0.000000


A0 = 1.495.17 A1 = 1.577.00 A2 = 1.571.83 A3 = 1.527.50 A4 = 1.787.50 A5 = 1.437.50



= 9.396,50 m2

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F. PENAMPANG 8-8’

1. Nilai Segmen


0 0 65 1 25 60 2 50 60 3 75 55 4 100 60 5 125 60 6 150 45

2. Diketahui

a0 = 65.00 a1 = 60.00 a2 = 60.00 a3 = 55.00 a4 = 60.00 a5 = 60.00 a6 = 45.00



M21 = H21/H11 = 25.00

a21 baru = a21 - M21 (a11) = 25 - (25 x 1) = 0.00 a22 baru = a22 - M21 (a12) = 25 - (25 x 0) = 100.00 a23 baru = a23 - M21 (a13) = 25 - (25 x 0) = 25.00 A21 baru = A21 - M21 (A11) = 0,6 - (25 x 0) = 0.60

M32 = H32/H22 = 0.25

a32 baru = a32 - M32 (a22) = 25 - (0.25 x 100) = 0 a33 baru = a33 - M32 (a23) = 100 - (0.25 x 25) = 93.75


a34 baru = a34 - M32 (a24) = 25 - (0.25 x 25) = 25.00

A31 baru = A31 - M21 (A21) = 0,6 - (0.25 x0,6) = -0.75

M43 = H43/H33 = 0.25

a43 baru = a43 - M43 (a33) = 25 - (0.25 x 100) = 0 a44 baru = a44 - M43 (a34) = 100 - (0.25 x 25) = 93.75 a45 baru = a45 - M43 (a35) = 25 - (0.25 x 25) = 25.00 A41 baru = A41 - M43 (A31) = 1,2 - (0.25 x-0,6) = 1.35

M54 = H54/H44 = 0.25

a54 baru = a54 - M54 (a44) = 25 - (0.25 x 100) = 0 a55 baru = a55 - M54 (a45) = 100 - (0.25 x 25) = 93.75 a56 baru = a56 - M54 (a46) = 25 - (0.25 x 25) = 25.00 A51 baru = A51 - M54 (A41) = 0,6 - (0.25 x 1,2) = -0.90

Matriks baru


C5 = 0 93.75C4 + 25C5 =-0,90 C4 = -0.00960 93.75C3 + 25C4 = 1,35 C3 = 0.01696

93.75C2 + 25C3 = -0,75 C2 = -0.01252 93.75C1 + 25C2 = 0,60 C1 = 0.00913

C0 = 0


b0 = -0.2761 b1 = -0.05 b2 = -0.13 b3 = 0.00 b4 = 0.16 b5 = -0.60


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d0 = 0.000122 d1 = -0.000289 d2 = 0.000393 d3 = -0.000354 d4 = 0.000128 d5 = 0.000000


A0 = 1550.61 A1 = 1504.42 A2 = 1431.72 A3 = 1427.92 A4 = 1512.50 A5 = 1312.50



= 8.739,67 m2

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HASIL PERHITUNGAN

1. Perhitungan Luas Penampang Dengan Aplikasi Polinomial Pangkat Tiga

No PENAMPANG LUAS (meter2) 1 1-1' 8.950,11 2 2-2' 9.583,33 3 3-3' 10.658,67 4 4-4' 10.731,28 5 5-5' 11.546,94 6 6-6' 11.069,00 7 7-7' 9.396,30 8 8-8' 8.739,67

TOTAL 80.675,30

2. Perhitungan Cadangan

Perhitungan volume dengan metode End Area dilakukan terhadap dua penampang yang saling berdekatan. Kemudian hasil perhitungan volume dikalikan dengan faktor tonase batubara yaitu 1.3 ton/m3.

NO Penampang Jarak (m) Volume (meter3) Tonase (ton)

1 1-2’ 25 231.668,00 3.011.684,00

2 2-3’ 25 253.025,00 3.289.325,00

3 3-4’ 25 267.374,38 3.475.866,88

4 4-5’ 25 278.477,75 3.620.210,75

5 5-6’ 25 282.699,25 3.675.090,25

6 6-7’ 25 255.816,25 3.325.611,25

7 7-8’ 25 226.699,63 2.947.095,13

TOTAL 1.795.760,25 23.344.883,25

PETA DASAR

0 1 2 3 4 5 6 7 8 9 100

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POLINOMIAL pangkat 3

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Transcript of POLINOMIAL pangkat 3