Lee's parents plan to celebrate his 21st birthday. This special event will be held at a popular hotel, starting at 18:00.

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 $(h-5)$.

$$\text{Cost} = 4000 + 1250(h-5)$$

The formula is $\boxed{\text{Cost} = 4000 + 1250(h-5)}$.

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: $\boxed{\text{P = 5250, Q = 6500, R = 10250}}$.

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 $\boxed{\text{step graph}}$.

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) + $(R6.75 \times 120 \text{ guests})$

$$\text{Venue Cost} = R7\,250 + R810 = R8\,060$$

• Birthday cake: $R1\,250$
• DJ 5-Star: The party is from 18:00 until 01:30.

$$\text{Duration} = 01:30 - 18:00 = 7 \text{ hours and } 30 \text{ minutes}$$

Since DJ 5-Star charges "per hour or part thereof", 7 hours and 30 minutes counts as 8 hours.

$$\text{DJ 5-Star Cost} = R1\,000 \times 8 = R8\,000$$

• Total budgeted expenses:

$$\text{Total} = R18\,000 + R8\,060 + R1\,250 + R8\,000 = R35\,310$$

The total budgeted expenses for the party are $\boxed{\text{R35 310}}$.

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:

• $\boxed{\text{DJ Cool was unavailable on the date of the party.}}$ (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 $\boxed{\text{R35 310}}$.

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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.

$$\text{Probability} = \frac{\text{Number of functions with >250 guests}}{\text{Total number of functions}} = \frac{6}{16} = \frac{3}{8} = 0.375$$

The probability is $\boxed{\text{0.375}}$.

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.