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Matematika Lanjut

Mid Exam Graduate Program on Mining Engineering

Course : Advanced Mathematics (TA5101)

Data:

According to gold exploration on vein-typed deposit using vertical drilling, following information are obtained: drilling database (Table 1); due to vertical drilling the thickness of both veins need to be corrected; rockmass (wall rock) density is about 2.1 ton/m3; point (0,0,0) is set to be the local reference. The lateral extend of vein 1 is characterised by its outcrop from (500,-888) to (-600,910), while vein 2 by its outcrop from (600, -848) to (-800, 1448). The pre-mining topography is flat at 0 masl.

Table 1: Drilling Database

DH-ID East North

Depth of Vein 1

Depth of Vein 2

Thickness of Vein 1

Thickness of Vein

2

Assay of Vein 1

Assay of Vein 2

(m) (m) (masl) (masl) (m) (m) (gr/ton) (gr/ton)1 0 0 50.0 -100.0 13.9 12.8 13.21 3.272 0 50 13.2 -136.8 11.3 14.1 5.68 8.823 0 100 -23.5 -173.5 11.6 13.9 10.33 5.994 0 150 -60.3 -210.3 12.4 13.3 13.81 4.505 50 0 -10.3 -160.3 12.6 14.8 8.11 7.586 100 0 -70.6 -220.6 11.4 13.1 11.87 8.947 150 0 -130.9 -280.9 11.8 16.2 3.35 8.508 200 0 -191.2 -341.2 13.1 12.6 7.75 3.739 50 50 -47.1 -197.1 10.0 13.2 5.42 3.4310 100 100 -144.1 -294.1 14.2 16.3 11.62 3.8611 150 150 -241.2 -391.2 13.9 15.6 5.56 8.3012 200 200 -338.2 -488.2 14.8 13.8 12.26 2.9613 25 50 -16.9 -166.9 13.7 16.0 8.88 8.0514 50 25 -28.7 -178.7 10.7 14.2 6.55 5.0815 75 100 -114.0 -264.0 12.9 14.6 5.53 9.8216 100 75 -125.7 -275.7 11.3 12.7 6.42 8.6917 100 150 -180.9 -330.9 12.7 13.9 7.78 5.3718 150 100 -204.4 -354.4 12.3 14.8 3.63 5.4419 150 200 -277.9 -427.9 11.0 16.5 13.65 4.1020 200 150 -301.5 -451.5 11.4 13.4 5.47 8.88

Ishaq22114016

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Matematika Lanjut

Give:

1. Mathematical model (equation) of both veins !2. Mathematical model (equation) of true thickness of both veins !3. Description of both veins (parallel / acrosses / perpendicular) to each other ! 4. Orientation of both veins (dip value and dip direction) !5. Estimated gold resource and tailing down to level -300 masl for both veins !

Note :

Be patient and don’t be panic !, please be relaxed while you complete the task ! It is possible to assume things that are not given.

Ishaq22114016

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Matematika Lanjut

Answer

1. Mathematical model (equation) of both veins a. Mathematical Model Vein 1 (menggunakan Surfer)z=−1.21 x−0.735 y+50

b. Mathematical Model Vein 2 (menggunakan Surfer)z=−1.21 x−0.735 y−100

2. Orientation of both veins (dip value and dip direction) a. Dip value vein 1

Topograpi merupakan bidang datar [0, 0, 1]

Persamaan vein 1: z=−1.21 x−0.736 y+50

cosθ=[ 0 ,0 ,1 ] .[−1.21 ,−0.735 ,−1]

√(−1.21)2+(−0.736)2+(−1)2

cosθ=−1

√3=−0.577

θ=arc cos (−0.577 )=125.24

Maka dip value vein adalah 180o – 125.24o = 54.76o

b. Dip value vein 2

Topograpi merupakan bidang datar [0, 0, 1]

Persamaan vein 2: z=−1.21 x−0.735 y−100

cosθ=[ 0 ,0 ,1 ] .[−1.21 ,−0.735 ,−1]

√(−1.21)2+(−0.736)2+(−1)2

cosθ=−1

√3=−0.577

θ=arc cos (−0.577 )=125.24

Maka dip value vein adalah 180o – 125.24o = 54.76o

c. Persamaan dan Azimuth Strike dan Dip Direction

Perpotongan antara bidang vein dengan topograpi (z = 0)

z=−1.21 x−0.735 y+50

0=−1.21x−0.735 y+50

Azimuth strikeStrike Dip direction

α

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Matematika Lanjut

tanα=−1.210.735

=−1 .6 5

α=−58.78o=58.78o

Sehingga azimuth strike adalah N 328.78o E

Untuk dip direction adalah N 58.78o E

Persamaan dip direction: 0=0.735 x−1.21 y+50

3. Description of both veins (parallel / acrosses / perpendicular) to each other Untuk kedua vein memiliki nilai dip yang sama yaitu, 54.76o terhadap bidang

topograpi. Hal ini mengindikasikan bahwa kedua vein sejajar/paralel terhadap kedalamannya.

4. Mathematical model (equation) of true thickness of both veins

a. Persamaan thickness vein 1 (menggunakan Surfer)

Z=0.00234 x+0.00271 y+11.9

Sehingga untuk true thickness-nya dapat dicari dengan mengalikan persamaan terhadap sin (35.24)

Z=0.00234 x+0.00271 y+11.9{sin (35.24 ) }

Z=0.00234 x+0.00271 y+11.9(0.577)

Z=0.00135 x+0.00156 y+6.87

b. Persamaan thickness vein 2 (menggunakan Surfer)

z=0.00093 x+0.00361 y+13.91

Sehingga untuk true thickness-nya dapat dicari dengan mengalikan persamaan terhadap sin (35.24)

z=0.00093 x+0.00361 y+13.91{sin (35.24 ) }

z=0.00093 x+0.00361 y+13.91(0.577)

z=0.00054 x+0.00208 y+8.03

Zb

Za35.24o

54.76o

Za : persamaan thicknessZb : persamaan true tickness

sin (35.24 )=zbza

zb=za sin (35.24)

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Matematika Lanjut

5. Estimated gold resource and tailing down to level -300 masl for both veins a. For Vein 1 Persamaan strike, z = 0

y1=−1.65 x+68.03

Persamaan strike, z = - 300

y2=−1.65 x+476.19

Out crop [500, -888] to [-600, 910]

x1=−600 x2=500

Persamaan true thickness vein 1; Z=0.00135 x+0.00156 y+6.87

Persamaan bidang vein 1; z=−1.21 x−0.735 y+50

Persamaan assay vein 1; z=−0.0166 x+0.0178 y+8.41

Menentukan Tonase Batuan (ρ=2.1ton

m3¿

VB = Bidang vein x true thickness x ρ

VB = (−1.21 x−0.735 y+50)(0.00135 x+0.00156 y+6.87¿(2.1)

= −0.0034 x2−0.0025 y2+721.35 ton

Integrasi;

VB = ∫−600

500

∫−1.65 x+68.03

−1.65x+476.19

(−0.0034 x2−0.0025 y2+721.35 )√1+ (−1.21 )2+ (−0.735 )2dydx

VB = ∫−600

500

∫−1.65 x+68.03

−1.65x+476.19

(−0.0034 x2−0.0025 y2+721.35 )√3dydx

VB = ∫−600

500

∫−1.65 x+68.03

−1.65x+476.19

(−0.0059x2−0.0043 y2+1249.42 )dy dx

= ∫−600

500

−2.4 x2+414767.13dx

= - 608556157 ton = 608556157 ton

Menentukan Resource Gold

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Matematika Lanjut

R = VB . persamaan assay

= (−0.0034 x2−0.0025 y2+721.35¿(−0.0166 x+0.0178 y+8.41¿

= 5.6 x10−5 x3−4.45 x10−5 y3+6066.55 gram

Integrasi Resource Gold

R = ∫−600

500

∫−1.65 x+68.03

−1.65x+476.19

(5.6 x 10−5 x3−4.45 x10−5 y3+6066.55 )√1+(−1.21 )2+(−0.735 )2dy dx

= ∫−600

500

∫−1.65 x+68.03

−1.65x+476.19

(5.6 x 10−5 x3−4.45 x10−5 y3+6066.55 )√3dy dx

= ∫−600

500

∫−1.65 x+68.03

−1.65x+476.19

( 9.63x 10−5 x3−7.71x 10−5 y3+10507.57 )dy dx

= ∫−600

500

3939.58x 10−5 x3+3753122.28 dx

= 18515413160 gram = 18515.41 ton

b. For Vein 2 Persamaan strike, z = 0

y1=−1.65 x−136.05

Persamaan strike, z = - 300

y2=−1.65 x+272.11

Out crop [600, -848] to [-800, 1448]

x1=−800 x2=600

Persamaan true thickness vein 2; Z=0.00054 x+0.00208 y+8.03

Persamaan bidang vein 2; z=−1.21 x−0.735 y−100

Persamaan assay vein 2; z=0.001144 x−0.00683 y+6.723

Menentukan Tonase Batuan (ρ=2.1ton

m3¿

VB = Bidang vein x true thickness x ρ

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Matematika Lanjut

VB = (−1.21 x−0.735 y−100 )(0.00054 x+0.00208 y+8.03¿(2.1)

= −13.7 x10−4 x2−3.2x 10−3 y2−1686.3 ton

Integrasi Tonase Batuan;

VB = ∫−800

6 00

∫−1.65x−136.05

−1.65x +272.11

(−13.7 x10−4 x2−3.2x 10−3 y2−1686.3 )√1+(−1.21 )2+(−0.735 )2dy dx

VB = ∫−800

600

∫−1.65x−136.05

−1.65x+272.11

(−13.7 x10−4 x2−3.2x 10−3 y2−1686.3 )√3dy dx

VB = ∫−600

500

∫−1.65 x+68.03

−1.65x+476.19

(−23.73x 10−4 x2−5.5 x10−3 y2−2920.76 )dy dx

= ∫−600

500

−0.97 x2−1314532.45dx

= - 2718425430 ton = 2718425430 ton

Menentukan Resource Gold

R = VB . persamaan assay

= (−13.7 x10−4 x2−3.2x 10−3 y2−1686.3¿(0.001144 x−0.00683 y+6.723¿

= −1.57 x10−6 x3+2.2 x10−5 y3−11337 gram

Integrasi Resource Gold

R = ∫−800

600

∫−1.65x−136.05

−1.65x+272.11

(−1.57 x10−6 x3+2.2 x10−5 y3−11337 )√1+ (−1.21 )2+ (−0.735 )2dy dx

= ∫−800

600

∫−1.65x−136.05

−1.65x+272.11

(−1.57 x10−6 x3+2.2 x10−5 y3−11337 )√3dy dx

= ∫−800

600

∫−1.65x−136.05

−1.65x+272.11

(−2.72x 10−6 x3+3.8x 10−5 y3−1 9636.26 )dy dx

= ∫−800

6 00

−1110.2 x 10−6 x3−7751075.2dx

= -11917741360 gram = 11917.74 ton

Sehingga tonase total batuan Tonase = 608 556 157 ton + 2 718 425 430 ton = 3 326 981 587 ton

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Matematika Lanjut

Resource Gold totalResource = 18 515.41 ton + 11 917.74 ton = 30 433.15 ton

Tailing total = 3 326 981 587 ton - 30 433.15 ton = 3 326 951 154 ton