a theater wants to take in at least $2000 for a certain matinee. children's tickets cost $5 each and adult tickets cost $10 each. the theatre can seat up to 350 people. find five combinations of children and adult tickets that will make their goal.
To find five combinations of children and adult tickets that would make at least $2000, we can use a systematic approach. Here's how we can calculate it:
1. Assume X to be the number of children's tickets sold.
2. The remaining tickets sold would be (350 - X) adult tickets.
3. Calculate the total revenue generated from children's tickets: revenue_children = X * $5.
4. Calculate the total revenue generated from adult tickets: revenue_adults = (350 - X) * $10.
5. Calculate the total revenue: total_revenue = revenue_children + revenue_adults.
By using this approach, we need to find five different values of X that satisfy the condition total_revenue ≥ $2000.
Let's calculate the combinations:
Combination 1:
Let's start with X = 0 (no children's tickets sold).
revenue_children = 0 * $5 = $0
revenue_adults = (350 - 0) * $10 = $3500
total_revenue = $0 + $3500 = $3500
This combination yields a total revenue of $3500, which is above the goal of $2000.
Combination 2:
Let's try X = 25 (25 children's tickets sold).
revenue_children = 25 * $5 = $125
revenue_adults = (350 - 25) * $10 = $3250
total_revenue = $125 + $3250 = $3375
This combination yields a total revenue of $3375, which is above the goal of $2000.
Combination 3:
Let's try X = 50 (50 children's tickets sold).
revenue_children = 50 * $5 = $250
revenue_adults = (350 - 50) * $10 = $3000
total_revenue = $250 + $3000 = $3250
This combination yields a total revenue of $3250, which is above the goal of $2000.
Combination 4:
Let's try X = 75 (75 children's tickets sold).
revenue_children = 75 * $5 = $375
revenue_adults = (350 - 75) * $10 = $2750
total_revenue = $375 + $2750 = $3125
This combination yields a total revenue of $3125, which is above the goal of $2000.
Combination 5:
Let's try X = 100 (100 children's tickets sold).
revenue_children = 100 * $5 = $500
revenue_adults = (350 - 100) * $10 = $2500
total_revenue = $500 + $2500 = $3000
This combination yields a total revenue of $3000, which is above the goal of $2000.
Therefore, the five combinations of children and adult tickets that would make at least $2000 are:
1. Children's tickets: 0, Adult tickets: 350 (Total revenue: $3500)
2. Children's tickets: 25, Adult tickets: 325 (Total revenue: $3375)
3. Children's tickets: 50, Adult tickets: 300 (Total revenue: $3250)
4. Children's tickets: 75, Adult tickets: 275 (Total revenue: $3125)
5. Children's tickets: 100, Adult tickets: 250 (Total revenue: $3000)
To find five combinations of children and adult tickets that will make at least $2000, we need to consider the constraints of the theater's capacity as well as the ticket prices.
Let's use a systematic approach to explore the combinations:
Combination 1:
Let's assume all 350 seats are filled with children. Since the cost of a children's ticket is $5, the amount earned for this combination is 350 * $5 = $1750. In this case, we still need to earn an additional $2000 - $1750 = $250. Thus, this combination does not meet the goal.
Combination 2:
Now, let's consider a scenario where all 350 seats are filled with adults. The amount earned for this combination would be 350 * $10 = $3500. Since this exceeds the goal of $2000, it is a valid combination.
Combination 3:
Let's try a combination of 200 children and 150 adults. The amount earned from children's tickets would be 200 * $5 = $1000, and from adult tickets, it would be 150 * $10 = $1500. The total amount earned would be $1000 + $1500 = $2500, which is also above the goal of $2000.
Combination 4:
Now, let's consider a combination of 100 children and 250 adults. The amount earned from children's tickets would be 100 * $5 = $500, and from adult tickets, it would be 250 * $10 = $2500. The total amount earned would be $500 + $2500 = $3000, which is above the goal of $2000.
Combination 5:
Lastly, let's consider a combination of 300 children and 50 adults. The amount earned from children's tickets would be 300 * $5 = $1500, and from adult tickets, it would be 50 * $10 = $500. The total amount earned would be $1500 + $500 = $2000, which meets the goal exactly.
Therefore, five combinations that will each generate at least $2000 are:
1. All adults (350 adults)
2. 200 children and 150 adults
3. 100 children and 250 adults
4. 300 children and 50 adults
5. All children (350 children) (Though this combination does not meet the goal, we've included it for completeness.)
Note: There could be additional valid combinations. These are just five examples.
number of adults --- x
number of children -- 350-x
10x + 5(350-x) ≥ 2000
5x ≥ 250
x ≥ 50
so just pick any 3 values of x so that x ≥ 50 but ≤ 350
e.g. x = 60,
number of adults = 60
number of kids = 290
take in = 10(60) + 5(290) = 2050 , that works
you pick 2 more