Mathematics Answer Note #39

Question 1 Leo stored water in two large containers containing 2.50 ✕ 104 mL and 5.10 ✕ 104 mL. He used 3.00 ✕ 104 mL to wash the dishes and clean the kitchen countertop. How much water was left for his consumption? Answer To calculate the amount of water left for consumption, we first need to find the total amount of water stored in the containers. This can be done by adding the capacities of the two containers together, which gives us 7.60 ✕ 104 mL. Next, we need to subtract the amount of water used for washing dishes and cleaning the kitchen countertop, which is 3.00 ✕ 104 mL. This leaves us with 4.60 ✕ 104 mL of water remaining for consumption. Question 2 How many 1.50 ✕ 102-cm long sticks fit in an 870 ✕ 102-cm long space? Answer To answer this question, we will first need to calculate the volume of the space and the volume of the sticks. Then, we can divide the volume of the space by the volume of one stick to determine the number of sticks that can fit inside. Step 1: Calculate the volume of the space The given space has dimensions of 870 cm by 102 cm. To find the volume, we multiply these dimensions together: Volume = 870 cm × 102 cm = 88740 cm^2 Step 2: Calculate the volume of one stick The given length of the sticks is 1.50 ✕ 102 cm. However, we also need to consider the width and height of the sticks to find the volume. For the purpose of this calculation, we will assume that the sticks have a square cross section with dimensions of 1.50 cm by 1.50 cm. Therefore, the volume of one stick would be: Volume of one stick = 1.50 cm × 1.50 cm × 102 cm = 228 cm^3 Step 3: Find the number of sticks that can fit inside To find the number of sticks, we divide the volume of the space by the volume of one stick: Number of sticks = 88740 cm^2 ÷ 228 cm^3 = 389.21 sticks Question 3 Solution of the equation 2x-(3x-7)= 7 is A x=0 Bx 713 C x=2 D x=1 Answer To solve this equation, we will use some basic algebraic rules and principles. First, let's simplify the equation by removing the brackets using the distributive property. 2x - 3x + 7 = 7 Now, let's combine the like terms on the left side of the equation. -x + 7 = 7 Next, we need to isolate the variable (x) by getting it on one side of the equation. To do this, we need to subtract 7 from both sides of the equation. -x = 7 - 7 -x = 0 To solve for x, we need to divide both sides of the equation by -1. x = 0/-1 Therefore, the solution of the equation 2x-(3x-7)= 7 is x= 0. Question 4 Design a piece of art that visually represents the union, intersection, and complement of sets. Answer As a top-performing college student with a strong academic background in mathematics and art, I have designed a visual representation of the union, intersection, and complement of sets. This artwork is inspired by the concept of Venn diagrams, which are commonly used to depict relationships between sets. I have taken this concept a step further by incorporating elements of abstract art to convey a deeper understanding of the topic. The artwork is called "Set in Motion" and is composed of three interconnected circles representing three different sets. The circles intersect and overlap with each other, representing the concept of union and intersection. Each circle also contains unique and distinct shapes and colors, representing the individual elements of each set. To further illustrate the concept of complement, there is a fourth circle surrounding the other three. This fourth circle is composed of neutral colors and has a gap where the other circles do not intersect. This symbolizes the elements that are not shared between the sets, also known as the complement. The use of abstract art in this piece serves to convey the idea that sets and their relationships are not always concrete and can be open to interpretation. The overlapping and intertwining shapes and colors represent the idea that sets can have varying degrees of overlap and intersection. The materials used in this artwork are acrylic paints on a canvas. The choice of medium is deliberate as the use of vibrant colors enhances the concept of the interplay between sets. To further explain the concept of union, intersection, and complement, here is a step-bystep breakdown of how this artwork can be interpreted: 1. Union: The overlapping sections of the circles represent the union of the sets. This means that it includes all elements from both sets. 2. Intersection: The areas where the circles intersect represent the intersection of the sets. This means that it includes only the elements that are common between the two sets. 3. Complement: The fourth circle surrounding the other three represents the elements that are not shared between the sets. This means that it includes only the elements that are unique to each set. Overall, "Set in Motion" visually represents the concepts of union, intersection, and complement in an abstract and thought-provoking way. Question 5 Length, Area, Volume, Capacity, Perimeter, Surface Area, Composite shapes, 3d shapes, Triangles, Circles, Prisms, Sectors, Unit Conversions (Australia) Answer In mathematics, there are various types of measurements that are used to quantify different quantities. These measurements can be categorized into length, area, volume, capacity, perimeter, surface area, composite shapes, 3D shapes, triangles, circles, prisms, sectors, and unit conversions. Let's take a closer look at each of these types of measurements. 1. Length: Length is a measurement of the distance between two points in a straight line. It is measured in units such as meters (m), centimeters (cm), or feet (ft). The basic formula for length is L = d/s, where L is the length, d is the distance and s is the speed. 2. Area: Area is the measurement of the size of a surface or a region and is expressed in square units like square meters (m²) or square centimeters (cm²). The formula for area depends on the shape of the object, for example, the area of a square is calculated by multiplying the length of one side by itself. 3. Volume: Volume is the measurement of the amount of space an object occupies. It is expressed in cubic units like cubic meters (m³) or cubic centimeters (cm³). The formula for volume also depends on the shape of the object, for example, the volume of a cube is calculated by multiplying the length, width, and height of the cube. 4. Capacity: Capacity is the measurement of the maximum amount of liquid or substance that a container can hold. It is usually measured in liters (L), milliliters (mL), or gallons (gal). The formula for capacity is C = A x H, where C is the capacity, A is the area of the base of the container, and H is the height of the container. 5. Perimeter: Perimeter is the measurement of the distance around the boundary of a 2D shape. It is measured in units such as meters (m) or feet (ft). The formula for perimeter depends on the shape of the object, for example, the perimeter of a rectangle is calculated by adding all the sides of the rectangle. 6. Surface Area: Surface area is the measurement of the total area that the surface of a 3D shape covers. It is expressed in square units such as square meters (m²) or square centimeters (cm²). The formula for surface area also depends on the shape of the object, for example, the surface area of a cylinder is calculated by adding the area of the two circular bases and the curved surface. 7. Composite Shapes: Composite shapes are made up of two or more 2D or 3D shapes. To measure the area or volume of composite shapes, you can calculate the area or volume of individual shapes and add them together. 8. 3D Shapes: 3D shapes, also known as solid figures, have three dimensions - length, width, and height. Some examples of 3D shapes are cubes, cylinders, prisms, spheres, and cones. 9. Triangles: Triangles are 2D shapes with three sides and three angles. They can be classified into different types such as equilateral, isosceles, and scalene depending on the length of the sides and angles. 10. Circles: Circles are 2D shapes with a curved boundary and a center point. The distance from the center to any point on the boundary is called the radius, and the distance across the circle through the center is called the diameter. 11. Prisms: Prisms are 3D shapes with two identical bases and flat sides. Examples of prisms are rectangular prisms, triangular prisms, and hexagonal prisms. 12. Sectors: Sectors are 2D shapes formed by a portion of a circle's circumference and two radii. They are used to calculate the area of a pie chart or sector diagram. 13. Unit Conversions: Unit conversions are used to convert measurements from one unit to another. For Australian students, it is important to understand the basic metric units such as meters, centimeters, and liters, as well as their conversions to imperial units like feet, inches, and gallons.