The drama department at a middle school wants to determine the price to charge for tickets to a show . If the price is too low, there won't be enough money to cover expenses. If its too high, they may not get a big enough audience. The teacher estimates that the profit, P, in dollars per show, can be represented by `P=-(t-12)^2+100` , where t is the price of a ticket in dollars . When the profit is 0, the drama department breaks even . What is the lowest ticket price for which the department breaks even ?

Answer

To find the lowest ticket price at which the drama department breaks even, we start with the profit equation given:

\[ P = -(t - 12)^2 + 100 \]

Where \( P \) is the profit in dollars and \( t \) is the price of a ticket in dollars. The department breaks even when the profit \( P \) is equal to 0.

### Step 1: Set the profit equation to zero
To find the break-even point, we set the profit equation to zero:

\[ 0 = -(t - 12)^2 + 100 \]

### Step 2: Rearrange the equation
We can rearrange the equation to isolate the squared term:

\[ (t - 12)^2 = 100 \]

### Step 3: Take the square root of both sides
Next, we take the square root of both sides. Remember that taking the square root will yield both a positive and a negative solution:

\[ t - 12 = \pm 10 \]

### Step 4: Solve for \( t \)
Now we solve for \( t \) in both cases:

1. **Positive case:**
\[ t - 12 = 10 \]
\[ t = 10 + 12 \]
\[ t = 22 \]

2. **Negative case:**
\[ t - 12 = -10 \]
\[ t = -10 + 12 \]
\[ t = 2 \]

### Step 5: Identify the lowest ticket price
Now we have two possible ticket prices where the department breaks even: \( t = 22 \) and \( t = 2 \). The question asks for the lowest ticket price:

The lowest ticket price is:

\[ t = 2 \]

### Final Answer
The lowest ticket price for which the drama department breaks even is **$2**.

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