Question

The unit cost, in dollars, to produce tubs of ice cream is $14 and the fixed cost is $21420. The revenue function, in dollars, is

Find the profit function.



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

Answer 1

To solve this problem, we need to determine the profit function and find the smallest break-even point for the production of tubs of ice cream. We know the following:

- Unit cost (c) = $14 per tub
- Fixed cost (F) = $21,420

Let \( x \) represent the number of tubs of ice cream produced and sold.

### 1. Finding Revenue Function
The revenue function (R) can be expressed as:
\[
R(x) = p \cdot x
\]
Where \( p \) is the selling price per tub. Since the selling price is not provided, we will denote it as \( p \).

### 2. Finding Cost Function
The total cost (C) consists of fixed costs and variable costs:
\[
C(x) = F + c \cdot x = 21,420 + 14x
\]

### 3. Finding Profit Function
The profit (P) is defined as the revenue minus the total cost:
\[
P(x) = R(x) - C(x)
\]
Substituting in the functions we derived:
\[
P(x) = (p \cdot x) - (21,420 + 14x)
\]
Simplifying this gives:
\[
P(x) = px - 14x - 21,420
\]
\[
P(x) = (p - 14)x - 21,420
\]

### 4. Break-Even Point
The break-even point occurs when profit equals zero:
\[
P(x) = 0
\]
Setting the profit function to zero:
\[
0 = (p - 14)x - 21,420
\]
Solving for \( x \):
\[
(p - 14)x = 21,420
\]
\[
x = \frac{21,420}{p - 14}
\]

### 5. Finding the Smallest Break-Even Point
To find the smallest break-even point in terms of tubs of ice cream, we need the selling price \( p \) to be greater than 14 (because if \( p \leq 14 \), the company would not make any profit). If we assume any selling price greater than $14, we can find the break-even quantity for specific values. However, in the absence of a specific value for \( p \), we cannot calculate an exact numeric value for the break-even point.

If you can provide a specific selling price, we can substitute that into the break-even formula \( x = \frac{21,420}{p - 14} \) to find the corresponding break-even quantity.

In conclusion:
- **Profit Function:** \( P(x) = (p - 14)x - 21,420 \)
- **Smallest Break-Even Point:** \( x = \frac{21,420}{p - 14} \) (where \( p > 14 \))