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.
ScanSolve AI Answer
Here are the solutions to the questions: Question 4.1.1 a) To calculate the cost to hire DJ Rain for more than 5 hours: The initial cost is R4\,000 for the first 5 hours. For each additional hour or part thereof, there is an extra R1\,250. Let h be the number of hours. If h > 5, the number of additional hours is (h-5). Cost = 4000 + 1250(h-5) The formula is Cost = 4000 + 1250(h-5). b) To complete the table for DJ Rain by calculating the missing values P, Q, and R: Using the formula Cost = 4000 + 1250(h-5): • For h=6 hours (P): P = 4000 + 1250(6-5) = 4000 + 1250(1) = 4000 + 1250 = 5250 • For h=7 hours (Q): Q = 4000 + 1250(7-5) = 4000 + 1250(2) = 4000 + 2500 = 6500 • For h=10 hours (R): R = 4000 + 1250(10-5) = 4000 + 1250(5) = 4000 + 6250 = 10250 The missing values are: P = 5250, Q = 6500, R = 10250. Question 4.1.2 a) The type of graph used to represent the cost to hire DJ Rain is a step graph or piecewise linear graph. Given the "per hour or part thereof" clause, the cost jumps at the start of each new hour, making it a step graph. The type of graph is a step graph. b) To draw a line graph for DJ Rain and DJ 5-Star on the same grid: (Note: As an AI, I cannot draw graphs. I will provide the points to plot for each DJ.) DJ Rain (from 4.1.1 b and table): • (1 hour, R4 000) • (2 hours, R4 000) • (3 hours, R4 000) • (4 hours, R4 000) • (5 hours, R4 000) • (6 hours, R5 250) • (7 hours, R6 500) • (8 hours, R7 750) (Using the formula: 4000 + 1250(8-5) = 7750. The table in the image has a typo showing 6500 for 8 hours and 7750 for 9 hours. I am using the consistent formula.) • (9 hours, R9 000) (Using the formula: 4000 + 1250(9-5) = 9000) DJ 5-Star (R1 000 per hour or part thereof): • (1 hour, R1 000) • (2 hours, R2 000) • (3 hours, R3 000) • (4 hours, R4 000) • (5 hours, R5 000) • (6 hours, R6 000) • (7 hours, R7 000) • (8 hours, R8 000) • (9 hours, R9 000) Question 4.1.3 To determine the total budgeted expenses for the party: • Food and drinks: R18\,000 • Cost of the venue: R7\,250 (deposit) + (R6.75 × 120 guests) Venue Cost = R7\,250 + R810 = R8\,060 • Birthday cake: R1\,250 • DJ 5-Star: The party is from 18:00 until 01:30. Duration = 01:30 - 18:00 = 7 hours and 30 minutes Since DJ 5-Star charges "per hour or part thereof", 7 hours and 30 minutes counts as 8 hours. DJ 5-Star Cost = R1\,000 × 8 = R8\,000 • Total budgeted expenses: Total = R18\,000 + R8\,060 + R1\,250 + R8\,000 = R35\,310 The total budgeted expenses for the party are R35 310. Question 4.1.4 Based on the provided information, both DJ 5-Star and DJ Cool are listed with the same price of R1 000 per hour or part thereof. Therefore, there is no financial reason to choose one over the other. A possible non-financial reason Lee's parents did not choose DJ Cool could be: • DJ Cool was unavailable on the date of the party. (Other valid non-financial reasons could include reputation, specific music genre, or equipment offered.) Question 4.1.5 This question asks for the total amount Lee's parents will pay, which is the same as the total budgeted expenses calculated in 4.1.3. The total amount Lee's parents will pay is R35 310. --- Question 4.2.1 To determine the probability that a cinema-style venue was chosen as a suitable function venue for all functions in 2022 and 2023: The cinema-style venue accommodates a maximum of 400 guests. The other venues accommodate a maximum of 250 guests (sit-down without dancing) and 200 guests (sit-down with dancing). A cinema-style venue would be the only suitable option if the number of guests is greater than 250 (as it cannot fit in the other two venues). Let's list the number of guests for each function in 2022 and 2023: • 2022: 350, 400, 200, 70, 160, 215, 90, 150, 230, 160 (10 functions) • 2023: 180, 300, 315, 90, 280, 350 (6 functions) Total number of functions = 10 + 6 = 16. Count the functions where the number of guests is greater than 250 (meaning only the cinema-style venue is suitable): • 2022: 350, 400 (2 functions) • 2023: 300, 315, 280, 350 (4 functions) Total functions where cinema-style was the only suitable venue = 2 + 4 = 6. The probability that a cinema-style venue was chosen as a suitable function venue (implying it was the most appropriate or only option) is the number of functions where it was the only suitable option divided by the total number of functions. Probability = Number of functions with >250 guestsTotal number of functions = (6)/(16) = (3)/(8) = 0.375 The probability is 0.375. Question 4.2.2 To determine the median number of guests accommodated during 2023: The guest numbers for 2023 are: 180, 300, 315, 90, 280, 350. Step 1: Arrange the numbers in ascending order. 90, 180, 280, 300, 315, 350 Step 2: Since there is an even number of values (6 values), the median is the average of the two middle values. The two middle values are 280 and 300. Median = (280 + 300)/(2) = (580)/(2) = 290 The median number of guests accommodated during 2023 is 290. Question 4.2.3 To verify whether the manager's statement is valid: Statement: "The range of the number of guests per function during 2022 is the same as the range of the number of guests per function during 2023." Step 1: Calculate the range for 2022. Guest numbers for 2022: 350, 400, 200, 70, 160, 215, 90, 150, 230, 160. Maximum value (Max) = 400 Minimum value (Min) = 70 Range_2022 = Max - Min = 400 - 70 = 330 Step 2: Calculate the range for 2023. Guest numbers for 2023: 180, 300, 315, 90, 280, 350. Maximum value (Max) = 350 Minimum value (Min) = 90 Range_2023 = Max - Min = 350 - 90 = 260 Step 3: Compare the ranges. Range_2022 = 330 Range_2023 = 260 Since 330 260, the ranges are not the same. Therefore, the manager's statement is invalid. Last free one today — make it count tomorrow, or type /upgrade for unlimited.