Question

The unit cost, in dollars, to produce tubs of ice cream is $20 and the fixed cost is $15876. The revenue function, in dollars, is R(x)=-2x^2+442x

Find the profit function. P(x)=



At what quantity is the smallest break-even point?
dollars
scoops of ice cream
gallons of ice cream
tubs of ice cream

Answer 1

To find the profit function \( P(x) \), we first need to determine both the total cost function \( C(x) \) and the revenue function \( R(x) \).

1. **Calculating the Cost Function \( C(x) \)**:
- The variable cost per tub is $20, so for \( x \) tubs, the variable cost is \( 20x \).
- The fixed cost is $15,876.
- Therefore, the total cost function \( C(x) \) is:
\[
C(x) = \text{Fixed Cost} + \text{Variable Cost} = 15876 + 20x
\]

2. **Profit Function**:
- The profit function \( P(x) \) can be calculated by subtracting the cost function from the revenue function:
\[
P(x) = R(x) - C(x) = (-2x^2 + 442x) - (15876 + 20x)
\]
- Simplifying this gives:
\[
P(x) = -2x^2 + 442x - 15876 - 20x
\]
\[
P(x) = -2x^2 + 422x - 15876
\]

So, the profit function is:
\[
P(x) = -2x^2 + 422x - 15876
\]

3. **Finding the Break-Even Points**:
- To find the break-even points, we need to set the profit function equal to zero:
\[
-2x^2 + 422x - 15876 = 0
\]
- To solve this quadratic equation, we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = -2 \), \( b = 422 \), and \( c = -15876 \).

- Plugging in the values:
\[
x = \frac{-422 \pm \sqrt{422^2 - 4 \cdot (-2) \cdot (-15876)}}{2 \cdot (-2)}
\]
\[
x = \frac{-422 \pm \sqrt{178084 - 126976}}{-4}
\]
\[
x = \frac{-422 \pm \sqrt{5108}}{-4}
\]
\[
x = \frac{-422 \pm 71.4}{-4} \quad (\text{approximate value for } \sqrt{5108})
\]

- Now calculating the two potential values for \( x \):
1. Choosing the plus sign:
\[
x = \frac{-422 + 71.4}{-4} = \frac{-350.6}{-4} \approx 87.65
\]
2. Choosing the minus sign:
\[
x = \frac{-422 - 71.4}{-4} = \frac{-493.4}{-4} \approx 123.35
\]

- Since \( x \) represents the quantity of tubs of ice cream, we consider the integer values. The smallest break-even point will be at \( x \approx 88 \) (rounding up as you cannot sell a fraction of a tub).

**Conclusion**:
- Profit function:
\[
P(x) = -2x^2 + 422x - 15876
\]
- The smallest break-even point is approximately **88 tubs of ice cream**.