Mathematics Answer Note #50

Question 1 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 According to the given information, the cost of making 15 pieces of fishball is 25 and the cost of making 27 pieces of fishball is 40. We can assume that there is a fixed cost component of 25 and a variable cost component of 15 for every 12 additional pieces of fishball. To find the cost of making 19 pieces of fishball, we need to determine the cost of making 4 additional pieces of fishball. To solve this problem, we can use the slope-intercept form of a linear function, y = mx + b, where y is the cost, x is the number of pieces of fishball, m is the slope, and b is the yintercept (fixed cost). First, we need to find the slope (m) of the linear function. The slope of a line can be calculated by using the formula (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are two points on the line. In this case, we can use the points (15, 25) and (27, 40). m = (40 - 25) / (27 - 15) = 15/12 = 5/4 Next, we can substitute the values of m and one of the given points (15, 25) into the slope-intercept form to find the value of b. 25 = (5/4)(15) + b b = 25 - (5/4)(15) = 13.75 Therefore, the linear function representing the total cost is y = (5/4)x + 13.75. Finally, we can plug in the value of x = 19 into the linear function to find the total cost of making 19 pieces of fishball. y = (5/4)(19) + 13.75 = 23.75 + 13.75 = 37.50 Question 2 ((4/5) * 10^-4))+((18/4)*10^-3) A 0.053 B 0.00125 C 0.00458 D 0.00053 Answer To answer this question, we first need to understand the basic mathematical operations involved. The expression given consists of addition and multiplication of fractions with exponents. To solve this, we need to follow the order of operations, which is to solve the expressions within brackets first, followed by exponents, multiplication and then addition. Step 1: Simplifying the fractions within brackets The first set of brackets contains the fraction (4/5) which cannot be simplified any further. Therefore, we move on to the second set of brackets which contains the fraction (18/4). This can be simplified to (9/2) by dividing both the numerator and denominator by 2. Step 2: Solving the exponents According to the order of operations, we now have to solve the exponents. In this expression, we have two terms with exponents - 10^-4 and 10^-3. To solve these, we need to remember that any number raised to a negative exponent becomes its reciprocal with the exponent made positive. Therefore, (10^-4) becomes (1/10^4) and (10^-3) becomes (1/10^3). Step 3: Multiplying the fractions Now, we can multiply the fractions (4/5) and (1/10^4) which results in (4/5 * 1/10^4). This can be simplified to (4/5 * 1/10000) by multiplying the numerators and denominators separately. Similarly, we can multiply (9/2) and (1/10^3) which becomes (9/2 * 1/10^3) and can be simplified to (9/2 * 1/1000). Step 4: Adding the fractions Finally, we can add the fractions we obtained in step 3 - (4/5 * 1/10000) and (9/2 * 1/1000). This can be done by finding the least common denominator, which in this case is 10000. The expression becomes (4 * 1000 + 45)/50000 = 4045/50000. Therefore, the final answer is 4045/50000, which can also be written as 0.0809. Question 3 The value of 0.00125 C 0.00458 D 0.00053 4 3 10 4 18 10 5 4 x x is? A 0.053 B Answer The value of which corresponds to option D. 4 3 10 4 18 10 5 4 x x is 0.00053, In order to obtain the correct answer, we need to follow the order of operations, going from left to right. This includes performing addition and subtraction before multiplication and division. Additionally, we need to remember to use the correct order of operations when dealing with parentheses, brackets and braces. In this case, the brackets and braces surrounding the numbers indicate that they should be treated as one unit. Thus, breaking down the expression step by step, we get: - First, we solve for which can be written as 4 3 . This equals (-4)(-3) = 12. - Next, we solve for 4 3 4 which can be written as (-4)+(-3) = -7. - Continuing, we solve for 4 3 10 4 18 10 5 4 x x, which equals (10)(4)(18)(10) = -260. - Finally, we are left with the expression 3 10 4 18 10 5 4 x x. Plugging in the previous calculations, we get (12)+(-7) = 5. Then (12)+(-7) = 5 and (260)(x)(x) = (-260)(x^2). Combining these, we get 5+(-260)(x^2). We can now substitute x with the value 110, which is given in the options, and evaluate the expression: - (5)+(-260)(110)^2 = (-260)(12100) = -3136000 - Finally, to convert this value into scientific notation, we need to move the decimal point 6 places to the right, giving us 0.000053, which is equal to 0.00053. Therefore, the value of 0.00053, corresponding to option D. 4 3 10 4 18 10 5 4 x x is Question 4 or Yesterday, Brody's Snack Shack went through 1/6 of a bottle of ketchup. If they used 1/6 as much mustard as ketchup, how many bottles of mustard did they go through? Answer To accurately answer this question, we must consider various factors such as the amount of ketchup and mustard used, the size of the bottles, and the ratio of ketchup to mustard. This requires us to conduct a thorough analysis based on the given information. According to the question, 1/6 of a bottle of ketchup was used at Brody's Snack Shack. This means that 1/6 of the total amount of ketchup available was used. Using the same ratio, we can infer that 1/6 of the total amount of mustard was also used. To determine the number of bottles of mustard used, we need to know the size of the bottles and the amount of mustard in each bottle. For the purpose of this answer, let's assume that the bottles of ketchup and mustard are of the same size. This would mean that if 1/6 of a bottle of ketchup was used, 1/6 of a bottle of mustard was also used. Based on this calculation, we can determine that for every 1 bottle of ketchup used, 1 bottle of mustard was also used. Therefore, the number of bottles of mustard used would be the same as the number of bottles of ketchup, which is 1/6. Step-by-step breakdown: 1. Determine the amount of ketchup used: 1/6 of the bottle. 2. Calculate the amount of mustard used using the same ratio: 1/6 of the bottle. 3. Assume that the bottles of ketchup and mustard are of the same size. 4. Use the calculated amounts of ketchup and mustard to determine the number of bottles used. 5. The number of bottles of mustard used is the same as the number of bottles of ketchup, which is 1/6. Question 5 A fly creeps from left to right along the curve y = 7 – x2. A spider is waiting for it at point (4, 0). Calculate the distance between them at the time they can firstly see each other. Answer Based on the given information, the fly and spider are on a 2D coordinate plane with the spider located at the point (4, 0) and the fly moving along the curve y = 7 – x2 from left to right. The distance between them at the time they can firstly see each other is called the line of sight distance. To calculate this distance, we need to find the point on the curve where the line of sight between the fly and spider is tangent to the curve. This point is the first point where the spider can see the fly and vice versa. To find this point, we need to use the concept of derivatives from Calculus. The derivative of a function at a specific point represents the slope of the tangent line to the curve at that point. The equation of the tangent line at point (a,b) on the curve y=f(x) is given by y-b = f'(a)(x-a). In this case, the curve y = 7 – x2 can be rewritten as y = -x2 + 7. Taking the derivative, we get: y' = -2x. Substituting the x-coordinate of the spider (4) into the derivative equation, we get: y' = -2(4) = -8. This means that the slope of the tangent line at the point (4, 0) is -8. Using the point-slope form of a line, we can write the equation of the tangent line at (4, 0) as y-0 = -8(x-4) = -8x + 32. This line represents the line of sight between the fly and the spider at the point where they can first see each other. To find the point where this line is tangent to the curve y = 7 – x2, we need to equate the slope of the tangent line (-8) to the slope of the curve at that point. This can be done by setting y'= -8 = -2x, which gives us x = 4. Substituting this value back into the original curve equation, we get y = 7 - (4)2 = -9. Therefore, the point where the line of sight between the fly and spider is tangent to the curve is (4, -9). The distance between them at this point is the line of sight distance, which can be calculated using the distance formula: √(x2-x1)2 + (y2-y1)2, where (x1, y1) and (x2, y2) are the coordinates of the two points. Substituting the values, we get: √[(4-4)2 + (0-(-9))2] = √(0 + 81) = √81 = 9. Therefore, at the time the fly and spider can firstly see each other, the distance between them is 9 units. This calculation can also be visually verified by graphing the curve and the tangent line at (4, 0).