This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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4. c. The cost of hiring a taxi is given by the formula: C = 2.50d + 5, where C is the cost in Ghana cedis and d is the distance in kilometres. i. Calculate the cost of hiring a taxi for 15 km. (2 marks) Step 1: Substitute d=15 into the formula. C = 2.50(15) + 5 Step 2: Calculate the cost. C = 37.50 + 5 C = 42.50 The cost of hiring a taxi for 15 km is GHC 42.50. ii. Make d the subject of the formula. (2 marks) Step 1: Subtract 5 from both sides of the equation. C - 5 = 2.50d Step 2: Divide both sides by 2.50. d = (C - 5)/(2.50) The subject is d = (C - 5)/(2.50). iii. Find the distance travelled if the cost was GHC 35. (2 marks) Step 1: Substitute C=35 into the formula. 35 = 2.50d + 5 Step 2: Subtract 5 from both sides. 35 - 5 = 2.50d 30 = 2.50d Step 3: Divide by 2.50. d = (30)/(2.50) d = 12 The distance travelled is 12 km. 5. a. Solve the following inequalities and illustrate the solution on a number line: (4 marks) i. 3x - 5 > 10 Step 1: Add 5 to both sides. 3x > 10 + 5 3x > 15 Step 2: Divide by 3. x > (15)/(3) x > 5 The solution is x > 5. Illustration on a number line: Draw a number line. Place an open circle at 5 and draw an arrow extending to the right. ii. 2(x+1) 3x - 4 Step 1: Distribute the 2 on the left side. 2x + 2 3x - 4 Step 2: Subtract 2x from both sides. 2 3x - 2x - 4 2 x - 4 Step 3: Add 4 to both sides. 2 + 4 x 6 x This can also be written as x 6. The solution is x 6. Illustration on a number line: Draw a number line. Place a closed circle (or shaded dot) at 6 and draw an arrow extending to the right. 5. b. A mobile phone company charges a base fee of GHC 10 per month plus GHC 0.50 per minute of calls. i. Write an inequality to represent the monthly cost being less than GHC 50. (2 marks) Let m be the number of minutes of calls. The total cost is 10 + 0.50m. The inequality is 10 + 0.50m < 50. ii. Solve the inequality to find the maximum number of minutes a customer can talk. (3 marks) Step 1: Subtract 10 from both sides. 0.50m < 50 - 10 0.50m < 40 Step 2: Divide by 0.50. m < (40)/(0.50) m < 80 Since the number of minutes must be an integer, the maximum number of minutes a customer can talk is 79. The maximum number of minutes is 79 minutes. iii. Interpret your answer in the context of the problem. (2 marks) To keep the monthly cost strictly below GHC 50, a customer must use less than 80 minutes of call time. Therefore, the maximum whole number of minutes they can talk is 79 minutes. 5. c. The sum of three consecutive integers is greater than 24 but less than or equal to 33. i. Write an inequality to represent this situation. (2 marks) Let the first integer be n. The three consecutive integers are n, n+1, and n+2. Their sum is n + (n+1) + (n+2) = 3n + 3. The inequality is 24 < 3n + 3 33. ii. Solve the inequality to find the possible values of the integers. (3 marks) Step 1: Solve the left part of the inequality: 24 < 3n + 3. 24 - 3 < 3n 21 < 3n (21)/(3) < n 7 < n Step 2: Solve the right part of the inequality: 3n + 3 33. 3n 33 - 3 3n 30 n (30)/(3) n 10 Step 3: Combine the two inequalities. 7 < n 10 Since n must be an integer, the possible values for n are 8, 9, 10. The possible values for the first integer are 8, 9, 10. iii. List all possible sets of three consecutive integers that satisfy the condition. (2 marks) • If n=8, the integers are 8, 9, 10. (Sum = 27) • If n=9, the integers are 9, 10, 11. (Sum = 30) • If n=10, the integers are 10, 11, 12. (Sum = 33) The possible sets of integers are \8, 9, 10\, \9, 10, 11\, \10, 11, 12\. 6. a. Find the gradient of the straight line joining the following pairs of points: (4 marks) The formula for the gradient m between two points (x_1, y_1) and (x_2, y_2) is m = (y_2 - y_1)/(x_2 - x_1). i. P(2, 3) and Q(5, 9) Step 1: Identify the coordinates. x_1=2, y_1=3, x_2=5, y_2=9. Step 2: Substitute the values into the gradient formula. m = (9 - 3)/(5 - 2) m = (6)/(3) m = 2 The gradient is 2. ii. A(-1, 4) and B(3, -4) Step 1: Identify the coordinates. x_1=-1, y_1=4, x_2=3, y_2=-4. Step 2: Substitute the values into the gradient formula. m = (-4 - 4)/(3 - (-1)) m = (-8)/(3 + 1) m = (-8)/(4) m = -2 The gradient is -2. Got more? Send 'em.