Tickets for a community theater production cost `$6 each when bougth in advance and $8 each when bought at the door. The theater group's goal is at least $2,000 in ticket sales for opening night. The theate group sold 142 opening-night tickets in advance. What is the minimum number of tickets they need to sell at the door on opening night to make their goal

Options

A. 143
B. 144
C. 192
D. 250

Answer

To solve the problem step by step, we need to determine how many tickets need to be sold at the door to meet the theater group's goal of at least $2,000 in ticket sales.

### Step 1: Calculate the revenue from advance ticket sales.
The cost of each advance ticket is $6, and the theater group sold 142 tickets in advance.

\[ \text{Revenue from advance tickets} = \text{Number of advance tickets} \times \text{Cost per advance ticket} \]
\[ \text{Revenue from advance tickets} = 142 \times 6 = 852 \]

### Step 2: Set up the equation for total revenue.
Let \( x \) be the number of tickets sold at the door. The cost of each ticket sold at the door is $8.

The total revenue from tickets sold at the door is:
\[ \text{Revenue from door tickets} = x \times 8 \]

The total revenue from both advance and door tickets must be at least $2,000:
\[ \text{Total Revenue} = \text{Revenue from advance tickets} + \text{Revenue from door tickets} \]
\[ 852 + 8x \geq 2000 \]

### Step 3: Solve the inequality for \( x \).
Subtract 852 from both sides:
\[ 8x \geq 2000 - 852 \]
\[ 8x \geq 1148 \]

Now, divide both sides by 8:
\[ x \geq \frac{1148}{8} \]
\[ x \geq 143.5 \]

### Step 4: Determine the minimum whole number of tickets.
Since \( x \) must be a whole number, we round up from 143.5 to the next whole number:
\[ x \geq 144 \]

### Conclusion:
The minimum number of tickets that need to be sold at the door to meet the goal of at least $2,000 in ticket sales is **144**.

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