MTH103 TASK 3 ASSIGNMENT (DUE FRIDAY MAY 29 AT 12 NOON)
Name: Lance Ganaban
Student ID: 1130996
Instructions:
1. This Task is to be completed as an Assignment. Be careful to maintain its integrity.
2. There is no need to complete a Cover Sheet – when submitting, it is assumed you have taken the necessary
care and that you understand the consequences of submitting any material that is not your own work.
3. Please attempt all five questions in the spaces provided showing a reasonable amount of working.
4. Be logical, clear and precise in the presentation of your thinking.
5. When complete submit your attempt as a .pdf file to MTH103Task1@usc.edu.au .
__________________________________
Question 1.
(a) (3 marks) A series of measurements were made during an experiment to calculate the total energy of a
system. The values 𝑚 = 15.5, ℎ = 1.22 and 𝑣 = 2.50 combined with 𝑔 = 9.81 (all in their SI units) are needed
for the formula 𝐸 = 𝑚𝑔ℎ + 𝑚𝑣2. By entering your calculations directly into the table below, use your
knowledge of significant figures to calculate the reported value of the total energy 𝐸.
𝑚𝑔ℎ
Calculator output
185.5071
48.4375
185.51
48.44
Reported as
to
to
233.9446
233.945
to
1 sig fig
2 sig figs
𝑚𝑣2
𝐸
3 decimal places
b. The efficiency 𝐻 of a computer compilation is given by 𝐻=1/𝑞+𝑝(1−𝑞) where 𝑝 is the number of computer
processors. Find 𝑝 for 𝐻=0.66 and 𝑞=0.83.
H = 1 / q + p (1-q)
pH = 1 / q + (1-q)
p = 1 / q + (1-q) H
p = 1 / 0.83 + 0.17 (0.66)
p = 1.06135
MTH103 TASK 3 ASSIGNMENT (DUE FRIDAY MAY 29 AT 12 NOON)
(b) (4 marks) Using a drone, the widths of an area burned by a bush fire were measured at 500m intervals, as
shown in the following table. Determine the approximate area burned by the bushfire using Simpson’s Rule.
Distance
(km)
0.000 0.500
1.000
1.500
2.000
2.500
3.000
3.500
4.000
Width
(km)
1.330 2.243
4.507
6.200
2.595
0.665
2.843
1.512
0.406
MTH103 TASK 3 ASSIGNMENT (DUE FRIDAY MAY 29 AT 12 NOON)
𝑑𝑥
ℎ
≈ ( ) (𝑦1
3
+ 4(𝑠𝑢𝑚 𝑜𝑓 𝑜𝑑𝑑𝑠)
+ 2(𝑠𝑢𝑚 𝑜𝑓 𝑒𝑣𝑒𝑛𝑠)
+ 𝑦𝑙𝑎𝑠𝑡)
Dx ≈ 0.5/3 (1.330
+4(10.62)+2(9.945)+0.406)
= 10.6843km
Question 2.
(a) (4 marks) Triangle 𝐴𝐵𝐶 has side 𝑎 = 74.8, side 𝑏 = 87.5 and angle 𝐶 = 62.0°. Solve for the
third side 𝑐 and the other two angles.
MTH103 TASK 3 ASSIGNMENT (DUE FRIDAY MAY 29 AT 12 NOON)
Side a = 74.8
Side b = 87.5
Side c = ?
Angle a = ?
Angle b = ?
Angle c = 62 degrees
Side c = (a2 + b2)1/2
= (74.82 + 87.52)1/2
= 115.114
Angle a
(b)
(2 marks) Scientists can use a set of footprints to calculate an animal’s step angle,
which is a measure of walking efficiency. In the photograph, it is angle 𝐴𝐵𝐶. The closer the
angle is to 180 degrees, the more efficiently the animal walked. For the set of footprints
shown, 𝐴𝐶 = 104cm, 𝐵𝐶 = 67cm and 𝐴𝐵 = 65cm. Find the step angle.
(c) (4 marks) Determine the distinct roots of the polynomial 𝑥3 − 4𝑥2 − 7𝑥 + 10.
MTH103 TASK 3 ASSIGNMENT (DUE FRIDAY MAY 29 AT 12 NOON)
Question 3.
(a) (3 marks) Find the vector cross product for 3D vectors 𝒂 and 𝒃 when
𝒂 = 2𝒊 – 𝒋 + 𝒌 and 𝒃 = 𝒊 + 𝒋.
MTH103 TASK 3 ASSIGNMENT (DUE FRIDAY MAY 29 AT 12 NOON)
(b) (5 marks) Find the magnitude and direction of the resultant force for the three forces shown in the
diagram.
MTH103 TASK 3 ASSIGNMENT (DUE FRIDAY MAY 29 AT 12 NOON)
(c) (2 marks) Find a unit vector in the same direction as the vector 𝒉 = 4𝒊 + 3𝒌, and show that it has
magnitude 1.
Question 4.
(a) (1 mark) Which of the following is the greater angle measure: 150.0° or 2.540 radians?
150 degrees. It is because it is larger when turned into radians.
(b)
(2 marks) Calculate the arc length 𝐿 and the area of the sector when 𝜃 = 2.21 radians
and 𝑟 = 2.10𝑚.
Arc length
Sector area
MTH103 TASK 3 ASSIGNMENT (DUE FRIDAY MAY 29 AT 12 NOON)
(c) (3 marks) Given 𝑀 = [10 2], 𝑁 = [3] , calculate 3𝑀𝑁 and 𝑀−1.
2.5 0
1
(d) (4 marks) Solve the following system of equations using a matrix method:
𝑥 + 𝑦 + 𝑧 = 4, 3𝑥 + 3𝑦 = 12,2𝑥 − 5𝑧 = 4.
MTH103 TASK 3 ASSIGNMENT (DUE FRIDAY MAY 29 AT 12 NOON)
Question 5.
(a) (2 marks) Steam in a boiler was heated to 150℃ and then allowed to cool. Its temperature 𝑇 was recorded
each minute as shown in the following table:
Time (min)
0.00
1.00
2.00
3.00
4.00
5.00
Temperature (℃)
150.0 142.8 138.5 135.2 132.7 130.8
Use linear interpolation to estimate the approximate temperature at 1 minute and 15 seconds.
(b) (8 marks) Solve the following equations for the unknown variable:
MTH103 TASK 3 ASSIGNMENT (DUE FRIDAY MAY 29 AT 12 NOON)
(i)
𝑥
−1 =
0.25
2𝑥
(ii)
cos 𝑥 = −0.866025 (180° ≤ 𝑥 ≤ 270°),
(iii)
𝑙𝑛 𝑐4.5 = 10.5
(to 2 decimal places)
(iv)
𝑒4𝑥+1 = 10𝑥
(to 4 decimal places).
(i)
(x-1)/2x = 0.25
x-1 = 0.25(2x)
-1 = 0.25 (x)
-1/0.25 = x
-4 = x
(iii)
(ii)
Cos x = -0.866025
X = cos-1(-0.866025)
X = 150
(iv)