ENGINEERING MATHEMATICS

Algebraic Methods



Algebraic Methods

1.

i)

ii)

For determining the constraint, regression analysis is applied.

Model Summaryb

Model

R

R Square

Adjusted R Square

Std. Error of the Estimate

Change Statistics

R Square Change

F Change

df1

df2

Sig. F Change

1

1.000a

1.000

1.000

.01314

1.000

85401.081

1

8

.000

a. Predictors: (Constant), logH

b. Dependent Variable: logQ

From the above model summary table, it is reflected that there is relationship between the log Q and logH, as the Adjusted R Square is 1, indicating strong relationship between the variables.

ANOVAa

Model

Sum of Squares

df

Mean Square

F

Sig.

1

Regression

14.756

1

14.756

85401.081

.000b

Residual

.001

8

.000

Total

14.758

9

a. Dependent Variable: logQ

b. Predictors: (Constant), logH

Moreover, the above table is showing that the regression model is statistically significant because the level of significance is below 0.05.

Coefficientsa

Model

Unstandardized Coefficients

Standardized Coefficients

t

Sig.

B

Std. Error

Beta

1

(Constant)

.314

.005

59.740

.000

logH

2.465

.008

1.000

292.235

.000

a. Dependent Variable: logQ

The coefficients table is showing that the independent variable is less than 0.05 showing the significance, which also indicates that there is positive relationship of logQ with the logH because the beta value is 2.465. Furthermore, the constant or the constraint in the regression model is 0.314. For that reason, k = 0.314 and n = 10.

iii)

Q = 0.314 H^10

iv)

Q = 0.314 (0.5^10)

Q = (0.314) (0.000976563)

Q = 0.000306641

v)

Q = 0.314 H^10

0.75 = 0.314 H^10

0.75 x 0.314 = H^10

0.2355 = H^10

Log 0.2355 = H^10

-1.446044366/ 10 = H

H = -0.144604437

2.

i)

P = 51200

f (V) = (P) * (0.5) / f (Y)

f (V) = (51200) * (0.5) / f (Y)

i.e. V = value of construction plant

Y = age of construction plant

ii)

As in 2 years, the value of the plant is 50% of its new cost; thus, after 4 years, the value of the plant will be 0.

iii)

f (V) = (P) * (0.5) / f (Y)

f (V) = (80000) * (0.5) / f (3)

f (V) = 13333

3.

i)

7x2 + 17x - 14

(x + 4) (2x2 + 3x - 5)

=

7x2

-

14 (2x2 + 3x - 5)

+

17x

x + 4

=

7x2

-

11 x

-

210

+

70

x + 4

ii)

3x2 - 20x + 30

(x - 4)2

3x2 - 20x

+

60

x - 4

x (3x - 20) +

60

x - 4

3x3 - 32x2 + 80x + 60

x - 4

4.

a)

Simplifying

5 = 4ln * 3x

Reorder the terms for easier multiplication:

5 = 4 * 3ln * x

Multiply 4 * 3

5 = 12ln * x

Multiply ln * x

5 = 12lnx

Solving

5 = 12lnx

Solving for variable “l”.

Move all terms containing l to the left, all other terms to the right.

Add '-12lnx' to each side of the equation.

5 + -12lnx = 12lnx + -12lnx

Combine like terms: 12lnx + -12lnx = 0

5 + -12lnx = 0

Add '-5' to each side of the equation.

5 + -5 + -12lnx = 0 + -5

Combine like terms: 5 + -5 = 0

0 + -12lnx = 0 + -5

-12lnx = 0 + -5

Combine like terms: 0 + -5 = -5

-12lnx = -5

x = - 5/ -121 ln

x = 5/ 121 ln

x = 0.4167/ ln

b)

Simplifying

2e3x = 7

Solving

2e3x = 7

Solving for variable 'e'.

Move all terms containing e to the left, all other terms to the right.

Divide each side by '2x'.

e3 = 3.5x-1

Simplifying

e3 = 3.5x-1

Combine like terms: 3.5x-1 + -3.5x-1 = 0.0

e3 + -3.5x-1 = 0.0

Factor out the Greatest Common Factor (GCF), 'x-1'.

x-1(e3x + -3.5) = 0.0

c)

3 coshx

=

27

x

=

27

3 cosh

x

=

9

cosh

d)

2 * sinh (3 * x) = 36.6

x

=

asinh

(18.3)

3

x

=

1.20026

5.

y

=

F

(

cosh

(

w

*

L

)

-

1

)

w

2

*

F

15

=

F

(

cosh

(

0.1

*

250

)

-

1

)

0.1

2

*

F

6.

v^2

=

1.8

? ...