Algebra foundations #1 - Questions and Answers

Question 1 A manufacturer of 24hr variable timers has a monthly fixed cost of 56000 and a production costs of 9 dollars for each timer manufactured the unit sell for 16 dollars each find the Break even point algebraically Answer 1 Step 1 Calculation: Cost function C(x) = 56000 + 9x. Revenue function R(x) = 16x. The unit sell for 16 dollars. Obtain the profit function as follows. Profit function P(x) = R(x) - C(x) = 16x - (56000 + 9x) = 16x-56000 - 9x =72-56000 Step 2 Solving the break even point to equate the profit function as zero. 7x-56000 = 0 7x = 56000 x= 56000 7 x = 8000 Thus, the break even point is (8000,0) Step 3 Answer: The break even point is (8000, 0). Answer 2 If the units sell for $16, and cost $9 to produce; then the profit on each unit is $7. Let n be the number of units needed to break even. Then: 7n = 56000 n = 8000 or (169)n - 56000 = 0 So, the break-even point is 8000 units; and the break-even revenue is $12800 Answer 3 The break even point occurs when the income from sales equals the cost to make the product Let x = number of timers cost = 5000 + 8x Income = 13x set them equal to find the break even point 5000+8x = 13x 5000 = 5x x = 1000(so 1000 timers must be sold each month to break even Profit = Income - cost Profit = 0 at the break even point So that is the point where it crosses the x axis (1000, 0) Question 2 To convert: 3.6 L into milliliters form. Answer Step 1 1ml = 0.001liter To convert from liters to milliliters, multiply the value of liter by Use the fraction method. 3.6L 3.6L = 1 3.6L Multiply the fraction 3.6L 1 3.6L 1 3.6L 1000ml X 1 1L 1000ml 1L = 3.6 x 1000ml = 3600ml Thus, 3600 ml is the conversion of 3.6 L. Question 3 Determine whether each equation represents a proportional relationship. If it does, identify the constant of proportionality. a. y = 0.5x - 2 b. y = 1,000x c. y = x + 1 Answer 1 An equation has a proportional relationship if it can be written in the form y=kx where k is the constant of proportionality. a. y = 0.5x-2 does not have a proportional relationship since it has the term -2 on the right side. b. y = 1,000z has a proportional relationship with a constant of proportionality of 1,000. c. y = x + 1 does not have a proportional relationship since it has the term 1 on the right side. Answer 2 Result: a) Equation y = 0.5x 2 does represent a proportional relationship with a constant of proportionality k = 0.5. b) Equation y = 1000x represents a proportional relationship with a constant of proportionality k = 1000. c) Equation y = x + 1 does not represent a proportional relationship. Solution: a) Equation: y = 0.5x - 2 To determine if this equation represents a proportional relationship, we can rewrite it in the form y = kx. 0.5x2 kx Rearranging the terms: 0.5xkx = 2 Factoring out the common factor of x: x(0.5-k) = 2 For this equation to hold true for all values of x, the expression in the parentheses, (0.5 – k), must be equal to zero. Therefore, k = 0.5 for this equation to represent a proportional relationship. b) Equation: y = 1000x This equation is already in the form y = kx. The constant of proportionality is k = 1000. c) Equation: y = x + 1 Similar to the previous equation, this equation is not in the form y = kx. The presence of the constant term 1 on the right side indicates that it does not represent a proportional relationship. Answer 3 We may compare the supplied equations to the general form of a proportional connection, which is y = kx, where k denotes the constant of proportionality, to determine whether each equation reflects a proportional relationship and to find the constant of proportionality. a) The equation y = 0.5x - 2 does not represent a proportional relationship because the coefficient of x is not a constant. It is 0.5, which means the ratio between y and x is not constant. Hence, we cannot identify a constant of proportionality for this equation. b) The equation y = 1,000x represents a proportional relationship. The coefficient of x is a constant, which is 1, 000. Therefore, the constant of proportionality for this equation is 1,000. c) The equation y = x + 1 does not represent a proportional relationship because it is in the form of a linear equation with a constant term (1) added to x. In a proportional relationship, there should not be a constant term added or subtracted. Therefore, we cannot identify a constant of proportionality for this equation. Answer: a) y = 0.5x - 2 does not represent a proportional relationship. No constant of proportionality. b) y = 1,000x represents a proportional relationship with a constant of proportionality of 1,000. c) y = x + 1 does not represent a proportional relationship. No constant of proportionality. Question 4 To convert: 1.9 L into milliliters form. Answer Step 1 Use the unit fraction method. 1.9L 1.9L = 1 Multiply the fraction 1.9L 1 1.9L = 1.9L 1000ml X 1 1L 1.9L 1000ml by 1 1L = 1.9 x 1000ml = 1900ml Thus, 1900ml is the conversion of 1.9 L Question 5 Write an algebraic expression of the verbal expression. 18 decreased by 3 times d Answer The word, «decreased by», includes a subtraction operation, which can be represented by the symbol <<-». The give expression indicates that <3 times d» is subtracted or deducted from 18 as shown below: 18-<3 times d>>> The word, <«times», is indicates a multiplication expression. The phrase, «3 times d», can be represented by the algebraic expression <3d». Therefore, we can express the entire verbal expression as the algebraic expression: 18-3d