Mathematics Answer Note #49

Question 1 Determine whether each fraction is a terminating or repeating decimal. Drop the fraction in the correct box in the answer columns. Terminating Decimal Repeating Decimal DRAG & DROP THE ANSWER 67\frac{6}{7} 7 6 178\frac{17}{8} 8 17 713\frac{7}{13} 13 7 3416\frac{34}{16} 16 34 Answer To determine whether a fraction is a terminating or repeating decimal, we must first understand what these terms mean. A terminating decimal is a decimal that ends after a finite number of digits, while a repeating decimal continues on with a repeating pattern of digits. To solve this question, we can use the concept of long division. By dividing the numerator by the denominator, we can determine whether the decimal will terminate or repeat. For 67\frac{6}{7}, we can divide 6 by 7 to get 0.857142857… The digit 6 does not continue to repeat, so this decimal is a terminating decimal. For 178\frac{17}{8}, we can divide 17 by 8 to get 2.125. This decimal also ends after a finite number of digits, making it a terminating decimal. For 713\frac{7}{13}, we can divide 7 by 13 to get 0.538461538… The digit 3 repeats, indicating that this is a repeating decimal. Lastly, for 3416\frac{34}{16}, we can divide 34 by 16 to get 2.125. This is similar to the previous fraction, 178\frac{17}{8}, which also results in a terminating decimal. Overall, we can see that the first two fractions result in terminating decimals, while the last two fractions result in repeating decimals. It is important to note that repeating decimals can also be written using a bar notation, such as 0.538461 repeating as 0.538overline{461}. Question 2 There are approximately 3.79 liters in one gallon. Kate has three, 3-Liter bottles of juice to mix together for punch. How many gallons of punch will this make? Answer To accurately determine how many gallons of punch Kate's three bottles of 3 liters each will make, we first need to convert the total amount of juice in liters to gallons. Then we can divide that number by the conversion factor of 3.79 liters per gallon to get the final answer. Step-by-Step Calculation: 1. Determine the total amount of juice in liters by multiplying the number of bottles (3) by the volume of each bottle (3 liters). 3 x 3 = 9 liters of juice in total 2. Convert the liters to gallons: 9 liters / 3.79 liters per gallon = 2.37 gallons Therefore, Kate's three, 3-liter bottles of juice will make 2.37 gallons of punch. Question 3 What is the Conversion Factor? How is it determined? How often does it change? APA style Answer The Conversion Factor is a numerical value that is used to convert a quantity from one unit to another. It is an essential factor in unit conversion, as it allows for consistency and accuracy when dealing with measurements. The Conversion Factor is determined by the relationship between the two units being converted, and it is usually a ratio between the two units. The most common way to determine the Conversion Factor is by using a conversion factor table or by performing a dimensional analysis. Dimensional analysis, also known as factor-label method, is a mathematical approach to unit conversion that involves multiplying the quantity by a conversion factor, which is essentially a fraction of equivalent units. This method is widely used in scientific fields and is considered the most accurate way of converting units. It involves breaking down the given quantity into its base units and then using the conversion factors to adjust the units to the desired form. For example, if we want to convert 10 inches to centimeters, we can use the conversion factor 2.54 cm/1 inch. Therefore, 10 inches is equivalent to 25.4 centimeters (10 inches x 2.54 cm/1 inch). Conversion factor tables, on the other hand, are pre-determined values that are used to convert units. These tables provide a quick and easy way to convert units without having to perform calculations. They can be found in textbooks, reference books, and online resources. For instance, the international system of units (SI) conversion factor table provides conversion values for length, mass, time, temperature, and other units. These tables are based on standard conversion factors and are constantly updated to ensure accuracy. The Conversion Factor is a critical aspect of unit conversion, and it is essential to note that it can change depending on the context. For instance, the conversion factor for converting temperature from Celsius to Fahrenheit is 1.8. However, this changes when converting from Fahrenheit to Celsius, where the conversion factor is 0.5556. Therefore, it is important to always double-check the conversion factor to ensure accuracy. In conclusion, the Conversion Factor is a crucial component in unit conversion, and it is determined by the relationship between the two units being converted. It can be calculated using dimensional analysis or referenced from conversion factor tables. It is also important to keep in mind that the conversion factor can vary depending on the units being converted. Therefore, it is crucial to always double-check and use the most updated conversion factor values for accurate results. Question 4 Given 4 2 6 3 f x x x , where one of the three roots is x 1 , which of the following statements is true? A The other two roots are real and unequal. B The other two roots are real and equal. C The other two roots are not real. D The graph of f x will not intersect the x -axis at all. Answer The answer to this question is option B: The other two roots are real and equal. This can be determined by using the discriminant formula for finding the roots of a quadratic function. The discriminant is given by the expression b^2-4ac, where a, b, and c are the coefficients of the quadratic function. In this case, a = 4, b = 2, and c = 6. Plugging these values into the discriminant formula, we get: b^2-4ac = 2^2-4(4)(6) = -32. Since the discriminant is negative, this means that the function has two complex roots. However, we are given that one root is x = 1, which is a real number. This means that the other two roots must be complex conjugates of each other, making them both real numbers. Question 5 In Beloy’s fish ball factory, it costs $25 to make 15 pieces of fishball while it costs $40 to make 27 pieces. Assuming that the total cost is described by a linear function, how much does it cost to make 19 pieces of fishball? A 30 B 31 C 32 D 33 Answer To answer this question accurately, we must understand what a linear function is and how it applies to this scenario. A linear function is a mathematical model that represents a straight line on a graph and shows the relationship between two variables. In this case, the variables are the number of fishball pieces and the cost of making them. Based on the given information, we can create a linear function to represent the total cost of making fishballs. Let us define the cost as C and the number of fishball pieces as N. We can then use the two given points, (15, $25) and (27, $40), to find the slope of the line, which is equal to the change in cost over the change in the number of fishball pieces. The slope can be calculated as (40-25)/(27-15) = $15/12 = $5/4. Now that we have the slope, we can use the point-slope form of a linear function to find the equation of the line, which is C - $25 = ($5/4)(N - 15). Simplifying this gives us C = ($5/4)N + $13.75. Using this equation, we can find the cost of making 19 fishball pieces by plugging in N = 19. C = ($5/4)(19) + $13.75 = $23.75. Therefore, the correct answer to this question is option B, $23.75. Why are the other answers wrong? Option A - Option A, $30, is incorrect because it does not consider the decreasing cost per unit as the number of fishball pieces increases. This means that the cost for 19 fishball pieces would be higher than $30. Option C - Option C, $32, is incorrect because it assumes a constant slope, which is not the case in this scenario. The slope of the linear function we calculated is $5/4, not a whole number. Option D - Option D, $33, is incorrect because it also assumes a constant slope of 1, which is not accurate as discussed in option C.