A movie theater has fixed costs of $5,500 per day and variable costs averaging $2 per customer. The theater charges $7 per ticket.
How many customers per day does the theater need in order to make a profit? Give the smallest possible value.
The theater makes a profit when it has more than _______ customers.
let n be the number of customers
7n - 2n ≥ 5500
5n ≥ 5500
n≥ 1100
Thanks
To determine the number of customers per day needed in order for the theater to make a profit, we need to calculate the total cost and total revenue.
Total cost per day = Fixed costs + (Variable costs per customer × Number of customers)
Total cost per day = $5,500 + ($2 × Number of customers)
Total revenue per day = Ticket price × Number of customers
Total revenue per day = $7 × Number of customers
For the theater to make a profit, the total revenue needs to be greater than the total cost.
Total revenue per day > Total cost per day
$7 × Number of customers > $5,500 + ($2 × Number of customers)
Simplifying the inequality:
7 × Number of customers > 5,500 + 2 × Number of customers
7 × Number of customers - 2 × Number of customers > 5,500
5 × Number of customers > 5,500
Number of customers > 5,500 / 5
The smallest possible value for the number of customers per day needed in order for the theater to make a profit is the next whole number greater than the result of the division.
Number of customers > 5,500 / 5
Number of customers > 1,100
Therefore, the theater makes a profit when it has more than 1,100 customers.
To find the number of customers per day the theater needs to make a profit, we need to consider the revenue and the costs.
The revenue per customer is $7, and the variable costs per customer is $2. This means that for each customer, the theater earns a profit of $7 - $2 = $5.
Now, let's calculate the daily revenue and the daily costs:
- Daily revenue = Revenue per customer × Number of customers per day
- Daily costs = Fixed costs per day + Variable costs per customer × Number of customers per day
The theater will make a profit when the daily revenue exceeds the daily costs:
Revenue per customer × Number of customers per day > Fixed costs per day + Variable costs per customer × Number of customers per day
Let's plug in the values given:
$5 × Number of customers per day > $5,500 + $2 × Number of customers per day
Simplifying the equation:
$5 × Number of customers per day - $2 × Number of customers per day > $5,500
Combining like terms:
$3 × Number of customers per day > $5,500
Now, to find the smallest whole number value of customers needed to make a profit, we divide both sides of the inequality by $3:
Number of customers per day > $5,500 / $3
Calculating the value:
Number of customers per day > 1,833.33
Since the theater can't have a fraction of a customer, we round up to the next whole number:
Number of customers per day > 1,834
Therefore, the theater makes a profit when it has more than 1,834 customers per day.