Question

The unit cost, in dollars, to produce tubs of ice cream is $10 and the fixed cost is $8370. The price-demand function, in dollars per tub, is p(x)=286-2x

Find the cost function.
C(x)=

Find the revenue function.
R(x)=


Find the profit function.
P(x)=

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 the given problem, we will find the cost function, revenue function, and profit function step by step, and determine the smallest break-even point.

### Cost Function, \( C(x) \)
The cost function is composed of the fixed costs and the variable costs. The variable cost per tub is given as $10 and the fixed cost is $8370.

\[
C(x) = \text{Fixed Cost} + \text{Variable Cost} \times \text{Quantity}
\]
\[
C(x) = 8370 + 10x
\]

### Revenue Function, \( R(x) \)
The revenue function is determined by the price per tub multiplied by the number of tubs sold. The price-demand function is given as \( p(x) = 286 - 2x \).

Thus, the revenue function is:
\[
R(x) = p(x) \times x = (286 - 2x) \times x
\]
\[
R(x) = 286x - 2x^2
\]

### Profit Function, \( P(x) \)
The profit function is the revenue function minus the cost function.
\[
P(x) = R(x) - C(x)
\]
Substituting the revenue and cost functions we calculated:
\[
P(x) = (286x - 2x^2) - (8370 + 10x)
\]
\[
P(x) = 286x - 2x^2 - 8370 - 10x
\]
\[
P(x) = -2x^2 + 276x - 8370
\]

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

\[
-2x^2 + 276x - 8370 = 0
\]

We can solve this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = -2 \), \( b = 276 \), and \( c = -8370 \).

Calculating the discriminant:

\[
b^2 - 4ac = 276^2 - 4(-2)(-8370) = 76176 - 66960 = 9216
\]

Now applying the quadratic formula:

\[
x = \frac{-276 \pm \sqrt{9216}}{2(-2)} = \frac{-276 \pm 96}{-4}
\]

Calculating the two possible values:

1. \( x = \frac{-276 + 96}{-4} = \frac{-180}{-4} = 45 \)
2. \( x = \frac{-276 - 96}{-4} = \frac{-372}{-4} = 93 \)

Thus, the break-even points occur at \( x = 45 \) tubs and \( x = 93 \) tubs.

The smallest break-even point is:
\[
\text{Smallest break-even point} = 45 \text{ tubs of ice cream}
\]

### Summary Outputs
- Cost function: \( C(x) = 8370 + 10x \)
- Revenue function: \( R(x) = 286x - 2x^2 \)
- Profit function: \( P(x) = -2x^2 + 276x - 8370 \)
- Smallest break-even point: 45 tubs of ice cream