Assignment Brief and Guidance:
Activity 01
Part 1
1. A tailor wants to make square shaped towels. The required squared pieces of cloth will be cut from a ream
of cloth which is 20 meters in length and 16 meters in width.
a. Find the minimum number of squared pieces that can be cut from the ream of cloth without wasting any
cloth.
b. Briefly explain the technique you used to solve (a).
2. On the first day of the month, 4 customers come to a restaurant. Afterwards, those 4 customers come to the
same restaurant once in 2,4,6 and 8 days respectively.
a. On which day of the month, will all the four customers come back to the restaurant together?
b. Briefly explain the technique you used to solve (a).
Part 2
3. Logs are stacked in a pile with 24 logs on the bottom row and 10 on the top row. There are 15 rows in all
with each row having one more log than the one above it.
a. How many logs are in the stack?
b. Briefly explain the technique you used to solve (a).
4. A company is offering a job with a salary of Rs. 50,000.00 for the first year and a 4% raise each year
after that. If that 4% raise continues every year,
a. Find the total amount of money an employee would earn in a 10-years career.
b. Briefly explain the technique you used to solve (a).
Part 3
5. Define the multiplicative inverse in modular arithmetic and identify the multiplicative inverse of 6 mod 13 while explaining the algorithm used.
6. Prime numbers are important to many fields. In the computing field also prime numbers are applied. Provide examples and in detail explain how prime numbers are important in the field of computing.
Activity 02
Part 1
1. Define 'Conditional Probability' with a suitable example.
2. The manager of a supermarket collected the data of 25 customers on a certain date. Out of them 5 purchased Biscuits, 10 purchased Milk, 8 purchased Fruits, 6 purchased both Milk and Fruits. Let B represents the randomly selected customer purchased Biscuits, M represents the randomly selected customer purchased Milk and F represents the randomly selected customer purchased Fruits.
Represent the given information in a Venn diagram. Use that Venn diagram to answer the following questions.
a. Find the probability that a randomly selected customer either purchased Biscuits or Milk
b. Show that the events “The randomly selected customer purchased Milk” and “The randomly selected
customer purchased Fruits” are independent.
3. Suppose a voter poll is taken in three states. Of the total population of the three states, 45% live in state A, 20% live in state B, and 35% live in state C. In state A, 40% of voters support the liberal candidate, in state B, 30% of the voters support the liberal candidate, and in state C, 60% of the voters support the liberal candidate.
Let A represents the event that voter is from state A, B represents the event that voter is from state B and
C represents the event that voter is from state C. Let L represents the event that a voter supports the
liberal candidate.
a. Find the probability that a randomly selected voter does not support the liberal candidate and lives
in state A.
b. Find the probability that a randomly selected voter supports the liberal candidate.
c. Given that a randomly selected voter supports the liberal candidate, find the probability that the
selected voter is from state B.
4. In a box, there are 4 types [Hearts, Clubs, Diamonds, Scorpions] of cards. There are 6 Hearts cards, 7
Clubs cards, 8 Diamonds cards and 5 Scorpions cards in the box. Two cards are selected randomly without
replacement.
a. Find the probability that the both selected cards are Hearts.
b. Find the probability that one card is Clubs and the other card is Diamonds.
c. Find the probability that the both selected cards are from the same type
Part 2
5. Differentiate between 'Discrete Random Variable' and 'Continuous Random Variable”.
6. Two fair cubes are rolled. The random variable X represents the difference between the values of the two
cubes.
a. Find the mean of this probability distribution. (i.e. Find E[X] )
b. Find the variance and standard deviation of this probability distribution. (i.e. Find V[X] and SD[X])
The random variables A and B are defined as follows:
A = X-10 and B = [(1/2)X]-5
c. Show that E[A] and E[B].
d. Find V[A] and V[B].
e. Arnold and Brian play a game using two fair cubes. The cubes are rolled, and Arnold records his score
using the random variable A and Brian uses the random variable B. They repeat this for a large number of
times and compare their scores. Comment on any likely differences or similarities of their scores.
7. A discrete random variable Y has the following probability distribution.
where k is a constant.
a. Find the value of k.
b. Find P(Y≤3).
c. Find P(Y>2).
Part 3
10. The “Titans” cricket team has a winning rate of 75%. The team is planning to play 10 matches in the next
season.
a. Let X be the number of matches that will be won by the team. What are the possible values of X?
b. What is the probability that the team will win exactly 6 matches?
c. What is the probability that the team will lose 2 or less matches?
d. What is the mean number of matches that the team will win?
e. What are the variance and the standard deviation of the number of matches that the team will win?
11. In a boys' school, there are 45 students in grade 10. The height of the students was measured. The mean height of the students was 154 cm and the standard deviation was 2 cm. Alex's height was 163 cm. Would his height be considered an outlier, if the height of the students were 11. In a boys' school, there are 45 students in grade 10. The height of the students was measured. The mean height of the students was 154 cm and the standard deviation was 2 cm. Alex's height was 163 cm. Would his height be considered an outlier, if the height of the students were normally distributed? Explain your answer.normally distributed? Explain your answer.
12. The battery life of a certain battery is normally distributed with a mean of 90 days and a standard
deviation of 3 days.
For each of the following questions, construct a normal distribution curve and provide the answer
a. About what percent of the products last between 87 and 93 days?
b. About what percent of the products last 84 or less days?
For each of the following questions, use the standard normal table and provide the answer.
c. About what percent of the products last between 89 and 94 days?
d. About what percent of the products last 95 or more days?
13. In the computing field, there are many applications of Probability theories. Hashing and Load Balancing are also included to those. Provide an example for an application of Probability in Hashing and an example for an application of Probability in Load Balancing. Then, evaluate in detail how Probability is used for each application while assessing the importance of using Probability to those applications
Activity 03
Part 1
1. Find the equation (formula) of a circle with radius r and center C(h,k) and if the Center of a circle is at (3,-1) and a point on the circle is (-2,1) find the formula of the circle.
2. Find the equation (formula) of a sphere with radius r and center C(h, k, l) and show that x^2 + y^2+ z^2 - 6x + 2y + 8z - 4 = 0 is an equation of a sphere. Also, find its center and radius.
3. Following figure shows a Parallelogram.
If a=(i+3j-k) , b=(7i-2j+4k), find the area of the Parallelogram
Part 2
4. If 2x - 4y =3, 5y = (-3)x + 10 are two functions. Evaluate the x, y values using graphical method.
5. Evaluate the surfaces in ℝ^3 that are represented by the following equations
i. y = 4
ii. z = 5
6. Following figure shows a Tetrahedron
Construct an equation to find the volume of the given Tetrahedron using vector methods and if the
vectors of the Tetrahedron are a=(i+4j-2k) , b=(3i-5j+k) and c=(-4i+3j+6k), find the volume of the
Tetrahedron using the above constructed equation..
Activity 04
Part 1
1. Determine the slope of the following functions.
i. f(x) = 2x - 3x^4 + 5x + 8
ii. f(x) = cos(2x) + 4x^2 - 3
2. Let the displacement function of a moving object is S(t) = 5t^3 - 3t^2 + 6t. What is the function for the velocity of the object at time t.
Part 2
3. Find the area between the two curves f(x) = 2x^2 + 1 and g(x) = 8 - 2x on the interval (-2) ≤ x ≤ 1 .
4. It is estimated that t years from now the tree plantation of a certain forest will be increasing at the rate of 3t^2 + 5t + 6 hundred trees per year. Environmentalists have found that the level of Oxygen in the forest increases at the rate of approximately 4 units per 100 trees. By how much will the Oxygen level in the forest increase during the next 3 years?
Part 3
5. Sketch the graph of f(x) = x^5 - 6x^3 + 3 by applying differentiation methods for analyzing where the graph is increasing/decreasing, local maximum/minimum points [Using the second derivative test], concave up/down intervals with inflection points.
6. Identify the maximum and minimum points of the function f(x)= 2x3 - 4x4 + 5x2 by further differentiation. [i.e Justify your answer using both first derivative test and second derivative test.
Sample Answer
Activity 01
Part 1
1.
a.
Length of the ream of the cloth is = 20m
Width of the ream of the cloth = 16m
Now we can find the area of the ream of the cloth.
The area of the ream of the cloth is = 20m*16m = 320 m²
Now let's find out the minimum number of squared pieces that can be cut from the ream of the cloth
without wasting any cloth
For that,
First, we must find Highest Common Factor of the length and width of the ream of the cloth.
So, now let's find the H.C.F of 20m and 16m.
20,16 H.C.F = 2+2 = 4
The H.C.F of 20m and 16m is 4m
Length and width of the square shaped towel is 4m.
Therefore
Area of the squared shaped towel is = 4m*4m =16m²
So, we must divide area of the ream of the cloth by area of the squared shaped towel. Therefore, can get
the minimum number of squared pieces that can be cut from the ream of the cloth without wasting any cloth.
320m÷16m = 20
minimum number of squared pieces that can be cut from the ream of the cloth without wasting any cloth is
20 pieces.
b. used ,Highest Common Factor (H.C.F) technique to solve this
Highest common factor (H.C.F) - highest common factor of two or more positive integers is the greatest
integer which divides each of them exactly. The great common devisor (G.C.F) also used for the H.C.F.
2.
a. Customers that come to a restaurant, On the first day of the month is = 4
Afterwards, those 4 customers come to the same restaurant once in = 2,4,6 and 8 days respectively.
divide the integers above by their prime factors. After, let's multiply those prime factors.
Now let's multiply these prime factors.
2 * 2 * 2 * 3 = 24
All the four customers come back to the restaurant together is on 24th day of the month.
b. I solve this using the least common multiplication (L.C.M.) technique.
Least common multiple (L.C.M) - An integer which is exactly divisible by two or more integers is called
a
common multiple of them.