Let's start by explicitly detailing each part of the problem.
### Step 1: Finding the Cost Function
The cost function represents the total cost of producing [tex]\( x \)[/tex] units of the product. It includes both fixed costs and variable costs:
- Fixed Cost (FC): This is the cost that does not change with the level of production, which is given as \[tex]$17284.
- Variable Cost (VC): This is the cost that changes with the level of production. It is given as \$[/tex]11 per unit, so if [tex]\( x \)[/tex] bins are produced, the variable cost is [tex]\( 11x \)[/tex].
Thus, the cost function [tex]\( C(x) \)[/tex] can be written as:
[tex]\[
C(x) = 11x + 17284
\][/tex]
### Step 2: Finding the Revenue Function
The revenue function represents the total income from selling [tex]\( x \)[/tex] units of the product. It is determined by multiplying the price-demand function [tex]\( p(x) \)[/tex] by the number of units [tex]\( x \)[/tex].
The price-demand function is given as:
[tex]\[
p(x) = 425 - 2x
\][/tex]
Therefore, the revenue function [tex]\( R(x) \)[/tex] is:
[tex]\[
R(x) = x \cdot p(x) = x \cdot (425 - 2x) = 425x - 2x^2
\][/tex]
### Step 3: Finding the Profit Function
The profit function represents the total profit from producing and selling [tex]\( x \)[/tex] units of the product. It is calculated as the difference between the revenue function and the cost function:
[tex]\[
P(x) = R(x) - C(x)
\][/tex]
Substituting [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex], we get:
[tex]\[
P(x) = (425x - 2x^2) - (11x + 17284)
\][/tex]
Simplifying this gives:
[tex]\[
P(x) = 425x - 2x^2 - 11x - 17284 = 414x - 2x^2 - 17284
\][/tex]
So, the profit function is:
[tex]\[
P(x) = -2x^2 + 414x - 17284
\][/tex]
### Step 4: Finding the Smallest Break-even Point
A break-even point occurs when the profit is zero, i.e., [tex]\( P(x) = 0 \)[/tex]. We solve the equation:
[tex]\[
-2x^2 + 414x - 17284 = 0
\][/tex]
Solving this quadratic equation for [tex]\( x \)[/tex], we find the break-even points. Of the solutions obtained, the smallest break-even point is:
[tex]\[
x = 58
\][/tex]
So, the smallest break-even quantity is:
[tex]\[
x = 58
\][/tex]
