Applied Mathematics For Civil Engineers
Contents
PART 13
Task 1:3
Task 2:5
PART210
Task 3:10
Task 4:13
PART 1
Task 1:
1) (a)
(b)
A frequency is a number near the in wave equation. Therefore, for the equation (a) the frequency is , and for the equation (b) it is .
The periodic time is equal to , where is frequency. So for the function (a) the periodic time is , and for the function (b) it is .
An amplitude is a number near the cosine or sine function. For the function (a) it is , and for the second one it is
When the two waves interact by superposition, they form the wave with the equation that equal the sum of their functions:
(c)
Using the compound angle formula
So
So
Pic.1 The red curve is the graph of the function , the green curve is , the blue curve is
The graphs of the functions and have the same frequency and period, but they have different amplitudes. They are periodic in x with period , and their amplitudes are 2 and corespondinlgy. The function starts from the origin.
The graph of the function has amplitude of , the same period , and it has the phase shift .
(d) (i)
(ii)
Task 2:
(2)
Gaussian elimination:
The first row is divided by 12.6
The second row is subtracted by the first row multiplied by 4.8; to the 3rd row is added the first row multiplied by 13
The second row is divided by
From the first row subtract the second row multiplied by ; from the 3rd row subtract the second row multiplied by
The third row is divided by
To the first row is added 3rd row multiplied by ; from the second row subtract the third row multiplied by
Cramer's method:
The answers are the same.
(3)
From the first equation
Thus,
Therefore,
The answer: (-3; -5); (5; 3)
(4)
The trigonometric Vieta’s method will be used
The chart in Excel
Pic.2 The graph of the function in Excel
Pic.3 Solving of the problem in Excel
PART2
Task 3:
(5) The perimeter is , and the area is
Thus,
is the function of a parabola. Since the coefficient of is negative, the parabola opens downward and a maximum is at the vertex point.
The formula for x coordinate of the vertex point , so . Therefore,
So the largest possible area is when the square is chosen with the side
The answer is
(6)
Total surface area of a cone,
The rate of change of surface area is
The answer is
(7)
Rotation around x axis
And then rotation around y axis
Let’s find limits
, but we also have to double this volume
(8)
a.
step
x
y
Local error
Global error
0
1
2
0
0
1
1.15
2.15
0.0097
0.0097
2
1.3
2.31957
0.0065
0.0150
3
1.45
2.5019
0.0045
0.0179
4
1.6
2.6931
0.0033
0.0193
5
1.75
2.89063
0.0025
0.0200
6
1.9
3.09286
0.0019
0.0203
7
2.05
3.2986
0.0015
0.0202
8
2.2
3.50732
0.0012
0.0199
9
2.35
3.71818
0.00098
0.0195
10
2.5
3.93085
0.00081
0.01914
b. From the graph and
Task 4:
(9)
a.
1. Find the binomial probabilities table with
2. Find the column containing
3. Find the probability when
From the table
b.
1. Find the binomial probabilities table with
2. Find the column containing
3. Find the probability when
From the table
c.
1. Find the binomial probabilities table with
2. Find the column containing
3. Find the probability when
From the table
(10)
a.
From the Poisson distribution table for the and ,
b.
X
Probability
0
1
2
3
4
5
6
7
8
c. The mean
The mean deviation
The variance
The standard deviation 0.004
(11)
The function of a regression line is , where
Using Excel the coefficients have been found
(12)
a.
Integrate both …
