Mathematics Answer Note #44

Question 1 I. Art Integrated Project Design a piece of art that visually represents the union, intersection, and complement of sets. Answer Designing a piece of art that visually represents the union, intersection, and complement of sets may seem like a daunting task, but with a thorough understanding of sets and their operations, it can be achieved. To begin, let us first define these key terms. A set is a well-defined collection of distinct objects, and elements of a set are represented within curly brackets { }. The union of two sets, denoted by the symbol ∪, is the combination of all elements from both sets without any repetition. The intersection of two sets, denoted by the symbol ∩, is the set of elements that are common to both sets. The complement of a set, denoted by the symbol ', is the set of all elements outside of a particular set. Now, to design a piece of art that visually represents these concepts, we can use the medium of Venn diagrams. Venn diagrams are used to visually depict the relationships between sets. They consist of overlapping circles, each representing a set, with the overlapping region representing the intersecting elements. To represent the union of sets A and B, we would draw two circles, one representing set A and the other representing set B. The overlapping region would represent the union of A and B. Similarly, to represent the intersection of sets A and B, we would draw two circles and the overlapping region would now represent the intersection of A and B. To visually represent the complement of a set A, we would draw one circle representing the universal set, which contains all possible elements, and subtract the circle representing set A from it. The remaining portion would represent the complement of set A. Other artistic ways to represent these set operations could include using different shapes and colors to represent the different sets and their relationships. For example, using squares to represent one set and circles to represent another, with overlapping or intersecting areas to show the set operations. Question 2 Elana can swim 12 laps in 4 minutes. Fill in the blanks in the double number line to show this relationship between laps and minutes Answer The given information implies a direct relationship between laps and minutes, specifically that for every 4 minutes of swimming, Elana can complete 12 laps. To represent this relationship on a double number line, we will use the concept of proportionality, where the ratio of laps to minutes remains constant. Firstly, we will label the two parallel lines on the number line as "Laps" and "Min" respectively. On the "Laps" line, we will mark the starting point at 0 and the endpoint at 12, representing the 12 laps completed by Elana in 4 minutes. Similarly, on the "Min" line, we will mark the starting point at 0 and the endpoint at 4, representing the 4 minutes taken by Elana to complete 12 laps. Next, we will divide each line into equal parts to represent the specific units. For the "Laps" line, we will divide it into 4 equal parts, each representing 3 laps. Similarly, for the "Min" line, we will divide it into 4 equal parts, each representing 1 minute. Finally, we will connect the corresponding points on the two lines with a straight line to show the relationship between the two quantities. This will result in a slope of 3, as 3 laps are completed for every 1 minute of swimming. It can also be represented as a ratio of 3 laps to 1 minute. Question 3 What is the value of x in the equation: 2x + 5 = 11 Can you solve for x? Answer To solve for x, we will need to use algebraic methods to isolate the variable x. Let's start by subtracting 5 from both sides of the equation since 5 and -5 are additive inverses, and it will cancel out on the left side of the equation. 2x + 5 - 5 = 11 - 5 2x = 6 Next, we need to get rid of the coefficient of 2 in front of x. To do this, we will divide both sides of the equation by 2, since division and multiplication are inverse operations. 2x/2 = 6/2 x=3 Therefore, the value of x in the given equation is 3. In summary, we used algebraic methods to isolate the variable x by subtracting 5 from both sides and dividing both sides by 2. This gave us the value of x, which is 3. Now, let's simplify this answer to its simplest form. Since 3 is already in its simplest form, there are no extra steps needed. Question 4 Find the area of a square with each side 35 milimeters Answer As a college student with top academic marks, I believe the best approach to finding the area of a square is by using the formula A = s², where A is the area and s is the length of one side. In this case, the length of each side is given as 35 millimeters. So, to find the area, we simply plug in the value of s into the formula as follows: A = (35 mm)² A = 35 mm x 35 mm A = 1225 mm² Therefore, the area of the square with each side measuring 35 millimeters is 1225 square millimeters. Question 5 A bag contains 3 purple, 4 green, 4blue, and 4 white marbles. What fraction of the marbles in the bag ate purple? Write as a fraction in simplest from Answer According to the given information, the bag contains a total of 3 purple, 4 green, 4 blue, and 4 white marbles. In order to determine the fraction of marbles that are purple, we need to divide the number of purple marbles by the total number of marbles in the bag. Thus, the fraction of purple marbles can be expressed as: 3 purple marbles/15 total marbles. This can be simplified by dividing both the numerator and denominator by the greatest common factor, which in this case is 3. Therefore, the simplified fraction for the purple marbles is: 1 purple marble/5 total marbles. We can also express this as a decimal and a percentage for better understanding. 1/5 = 0.2 = 20%. Hence, the fraction of marbles in the bag that are purple is 1/5 or 20%.