Question

The unit cost, in dollars, to produce bins of cat food is $5 and the fixed cost is $11352. The revenue function, in dollars, is R(x)=-2x^2+355x

Find the cost function.
C(x)=


Find the profit function.
P(x)=


At what quantity is the smallest break-even point?
dollars
bins of cat food
servings of cat food
grams of cat food

Answer 1

To determine the cost function \( C(x) \) and the profit function \( P(x) \), we start with the information provided:

1. **Unit Cost**: Each bin of cat food costs $5 to produce.
2. **Fixed Cost**: The fixed cost is $11,352.
3. **Revenue Function**: The revenue function is given as \( R(x) = -2x^2 + 355x \).

### Step 1: Finding the Cost Function \( C(x) \)

The total cost function \( C(x) \) consists of the fixed costs plus the variable costs, which depend on the number of bins produced (x).

The variable cost is calculated as the unit cost multiplied by the number of bins. Thus:

\[
C(x) = \text{Fixed Cost} + \text{Variable Cost} = 11352 + 5x
\]

So, the cost function is:

\[
C(x) = 11352 + 5x
\]

### Step 2: Finding the Profit Function \( P(x) \)

The profit function \( P(x) \) is defined as revenues minus costs:

\[
P(x) = R(x) - C(x)
\]

Substituting the expressions for \( R(x) \) and \( C(x) \):

\[
P(x) = (-2x^2 + 355x) - (11352 + 5x)
\]

Now, combine like terms:

\[
P(x) = -2x^2 + 355x - 11352 - 5x
\]
\[
P(x) = -2x^2 + 350x - 11352
\]

### Summary of Functions
- Cost function: \(\boxed{C(x) = 11352 + 5x}\)
- Profit function: \(\boxed{P(x) = -2x^2 + 350x - 11352}\)

### Step 3: Finding the Break-Even Point

To find the break-even point, we set the profit function \( P(x) \) equal to zero:

\[
-2x^2 + 350x - 11352 = 0
\]

We can solve this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -2 \), \( b = 350 \), and \( c = -11352 \).

Calculating the discriminant:

\[
b^2 - 4ac = 350^2 - 4(-2)(-11352)
\]
\[
= 122500 - 90816
\]
\[
= 31684
\]

Now using the quadratic formula:

\[
x = \frac{-350 \pm \sqrt{31684}}{2(-2)}
\]
\[
= \frac{-350 \pm 178.08}{-4}
\]
Calculating the two possible values for \( x \):

1.
\[
x_1 = \frac{-350 + 178.08}{-4} = \frac{-171.92}{-4} \approx 42.98 \quad (\text{approximately } 43)
\]

2.
\[
x_2 = \frac{-350 - 178.08}{-4} = \frac{-528.08}{-4} \approx 132.02 \quad (\text{approximately } 132)
\]

### Conclusion on Break-Even Points

The smallest break-even point occurs at approximately \( 43 \) bins of cat food.

**Final Answer:**
The smallest break-even point is \(\boxed{43}\) bins of cat food.