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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Here are the solutions to the problems you've provided: SECTION B: STRUCTURAL 1. A 70 seater passenger bus leaves Bamenda for Yaounde, a distance of 400km. The driver pays 5000 FCFA at each toll gate located after every 150km. a. Find the number of toll gates between the two towns. Step 1: Divide the total distance by the distance between toll gates. Number of toll gates = Total distanceDistance per gate Number of toll gates = 400 km150 km Number of toll gates = 2.66... Number of toll gates = 2 The number of toll gates is 2. b. The total amount paid for the toll gates. Step 1: Multiply the number of toll gates by the cost per gate. Total toll gate cost = Number of gates × Cost per gate Total toll gate cost = 2 × 5000 FCFA Total toll gate cost = 10000 FCFA The total amount paid for the toll gates is 10000 FCFA. c. The actual number of passengers carried, given that two seats are reserved for the driver and the bus conductor. Step 1: Subtract the reserved seats from the total seats. Actual passengers = Total seats - Reserved seats Actual passengers = 70 - 2 Actual passengers = 68 The actual number of passengers carried is 68. d. The fare is 5000 FCFA per passenger and the bus uses 100,000 FCFA for fuel. Calculate: i. The total amount collected for the fare. Step 1: Multiply the fare per passenger by the actual number of passengers. Total collected = Fare per passenger × Actual passengers Total collected = 5000 FCFA × 68 Total collected = 340000 FCFA The total amount collected for the fare is 340000 FCFA. ii. The balance after all the expenditures are made. Step 1: Calculate total expenditures. Total expenditures = Toll gate cost + Fuel cost Total expenditures = 10000 FCFA + 100000 FCFA Total expenditures = 110000 FCFA Step 2: Calculate the balance. Balance = Total collected - Total expenditures Balance = 340000 FCFA - 110000 FCFA Balance = 230000 FCFA The balance after all expenditures is 230000 FCFA. 2. Given the functions f and g, defined as f(x) = x-3, x R and g(x) = 2x+1. Find: a. f(4) Step 1: Substitute x=4 into the function f(x). f(4) = 4 - 3 f(4) = 1 The value of f(4) is 1. b. fg(x) Step 1: Substitute g(x) into f(x). fg(x) = f(g(x)) fg(x) = f(2x+1) Step 2: Apply the definition of f(x) to 2x+1. fg(x) = (2x+1) - 3 fg(x) = 2x - 2 The function fg(x) is 2x-2. c. g^-1(x) Step 1: Let y = g(x). y = 2x+1 Step 2: Swap x and y. x = 2y+1 Step 3: Solve for y. x - 1 = 2y y = (x-1)/(2) The inverse function g^-1(x) is (x-1)/(2). 3. Figure 1 is a Venn diagram representing the number of students in a class of 30; P = Students studying physics, C = Students studying chemistry. a. Write, in set notation, the total number of students in the class. The total number of students in the class is represented by the cardinality of the universal set. n(E) = 30 The total number of students in the class is n(E) = 30. b. Write down an equation in terms of x. The sum of students in all regions of the Venn diagram must equal the total number of students in the class. x + 4 + 5 + 2x = 30 The equation in terms of x is x + 4 + 5 + 2x = 30. c. Hence, find the value of x. Step 1: Combine like terms in the equation. 3x + 9 = 30 Step 2: Subtract 9 from both sides. 3x = 30 - 9 3x = 21 Step 3: Divide by 3. x = (21)/(3) x = 7 The value of x is 7. d. Find, how many students study neither physics nor chemistry. Step 1: From the Venn diagram, the number of students studying neither subject is 2x. Step 2: Substitute the value of x=7. Students studying neither = 2 × 7 Students studying neither = 14 The number of students who study neither physics nor chemistry is 14. e. Determine the number of students studying one subject only. Step 1: Students studying one subject only are those in P only or C only. Students studying one subject only = (Students in P only) + (Students in C only) Students studying one subject only = x + 5 Step 2: Substitute the value of x=7. Students studying one subject only = 7 + 5 Students studying one subject only = 12 The number of students studying one subject only is 12. Last free one today — make it count tomorrow, or type /upgrade for unlimited.