Mathematics Answer Note #4

Question 1 Make a project on cylinder in 10 pages of class 9 Answer Introduction: In this project, we will explore the concept of a cylinder, which is a three-dimensional shape with two circular bases. We will explore its properties, formulas, and real-life applications. Our aim is to gain a deeper understanding of this geometrical shape and its significance in different fields. Properties of a Cylinder: A cylinder has two bases, which are parallel and congruent circles. The distance between the two bases is called the height of the cylinder. The radius of the bases is a measure of the distance from the center of the circular base to its outer edge. It is worth noting that the radius is half the diameter of the base. The surface area and volume of a cylinder can be calculated using specific formulas, as discussed below. Formulas for a Cylinder: Surface Area of a Cylinder (SA) = 2πr(r+h) Volume of a Cylinder (V) = πr²h Where: SA- Surface Area V- Volume π –Pi (approximately 3.14) r- Radius of the base h- Height of the cylinder Step-by-Step Calculation Example: Let us consider a cylinder with a radius of 4cm and a height of 10cm. Surface Area (SA) = 2π(4)(4+10) SA = 2π(4)(14) SA = 112π cm² Volume (V) = π(4)²(10) cm³ V = π(16)(10) cm³ V = 160π cm³ Real-Life Applications: Cylinders have a variety of real-life applications, some of which are mentioned below: 1. Canisters and containers: Cylinders are commonly used to make storage containers and canisters for holding liquids, gases, or other substances. For example, gas cylinders used for storing propane or oxygen. 2. Hydraulic Cylinders: In engineering, hydraulic cylinders are used to convert the energy of pressurized fluids into motion. These are commonly used in heavy machinery, like cranes, bulldozers, etc. 3. Pipes and Tubes: Cylindrical pipes or tubes are widely used for plumbing, water transportation, and construction purposes. 4. Rainfall Measurement: Rain gauges, which are used for measuring rainfall, are often shaped like a cylinder. 5. Musical Instruments: Some musical instruments, like the flute, are made in the shape of a cylinder. Conclusion: In conclusion, cylinders are an essential shape in geometry. They are widely used in different industries and have various real-life applications. In this project, we have explored the properties of a cylinder, its formulas, and some of its practical uses. We hope this project has helped in understanding this geometrical shape better and its significance in our daily lives. Question 2 Input principle, rate and time . Calculate and print simple interest Answer To calculate the simple interest, we need to understand the basic formula: Simple Interest = (Principle * Rate * Time)/100. Before we dive into the calculation, let's define the terms used in this formula. "Principle" refers to the initial amount of money borrowed or invested, "rate" is the percentage of interest charged, and "time" is the total time period in years. Simple interest is the interest paid only on the principal amount and does not include any interest on interest. Now, let's take an example to understand the calculation better. Suppose, we borrow $500 at an interest rate of 5% for a period of 3 years. Using the formula, we can calculate the simple interest as follows: Simple Interest = (500 * 5 * 3)/100 = $75 Therefore, the total amount we need to repay at the end of 3 years will be $575 (principal amount + simple interest). Question 3 If a yard is three feet long and there are 5,280 feet in 1 mile, how many miles must our football field be to fit this star into our model? Answer To answer this question, we must first understand the units of measurement used. A yard is a unit of length which is equivalent to three feet. On the other hand, a mile is a longer unit of length that is equal to 5,280 feet. Therefore, to convert from yards to miles, we must divide by 5,280. To fit a star into our football field model, we need to determine the length of our football field in miles. Given that we know that a yard is three feet long, we can use this information to find the length of the football field in yards. Depending on the size of the star and the model of the football field, we can then calculate the appropriate number of miles needed. To calculate the length of our football field in yards, we can use the formula: length (in yards) = length (in feet) / 3. Since the length of our football field is unknown, let's use the variable 'x' to represent it. Therefore, we can write the equation as: x (yards) = x (feet) / 3. Now, we can plug in the value of x (feet) by multiplying the length of the football field (in feet) by the number of stars we intend to fit in the field. For example, if our football field is 100 feet long and we want to fit three stars in it, then x (feet) = 100 feet x 3 stars = 300 feet. Substituting the value of x (feet) into our equation, we get: x (yards) = 300 feet / 3 = 100 yards. This means that our football field needs to be 100 yards in length to fit three stars into the model. To convert yards to miles, we divide by 5,280. Thus, our football field needs to be 100 yards / 5,280 = 0.0189393939 miles long to fit three stars into the model. Question 4 Property taxes are the product of the tax rate (T) and the assessed value (V). The total property tax collected in your city (P) is: P = T•V. If the value of properties rises 4\text% and if Mayor and City Council reduces the property the tax rate by 2%, What happens to the total amount of property tax collected? (Hint: the percentage rate of change of a product is approximately the sum of the percentage rates of change.) It rises 6% It rises 4% It rises 3% It rises 2% It falls 2% Answer To understand the impact of the change in property values and tax rate on the total amount of property tax collected in the city, we must first understand the formula for calculating property tax. As stated in the question, property tax is the product of the tax rate and the assessed value, represented by the equation P = T•V. This means that the total property tax collected (P) is dependent on both the tax rate (T) and the assessed value (V). In order to calculate the percentage change in the total property tax collected, we must consider the percentage changes in the individual components of the formula - the tax rate and the assessed value. The formula for calculating the percentage change of a product is approximately the sum of the percentage rates of change (Source: Study.com). Thus, in this scenario, the percentage change in the total property tax collected can be calculated as follows: Total percentage change = Percentage change in assessed value + Percentage change in tax rate = (4%) + (-2%) // Since the assessed value has risen by 4% and the tax rate has been reduced by 2% = 2% Therefore, the total amount of property tax collected in the city will rise by 2%. This means that the city will collect 2% more property tax revenue than before. Question 5 Given that f, of, x, equals, x, squared, plus, 2, x, minus, 8f(x)=x 2 +2x−8 and g, of, x, equals, x, minus, 2g(x)=x−2, find left parenthesis, f, plus, g, right parenthesis, left parenthesis, x, right parenthesis(f+g)(x) and express the result as a polynomial in simplest form. Answer To solve this problem, we first need to understand what (f+g)(x) means. (f+g)(x) represents the sum of the functions f(x) and g(x), or in other words, adding the two functions together. Since both f(x) and g(x) are polynomials, we can simply combine like terms to find the sum. (f+g)(x) = f(x) + g(x) // definition of (f+g)(x) = (x^2 + 2x - 8) + (x - 2) // substituting the given functions = x^2 + (2x + x) + (-8 - 2) // combining like terms = x^2 + 3x - 10 // simplified form Hence, the result of (f+g)(x) is a polynomial of degree 2 with a leading coefficient of 1 and the constants 3 and -10.