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The unit cost, in dollars, to produce bins of cat food is [tex] \$11 [/tex] and the fixed cost is [tex] \$17284 [/tex]. The price-demand function, in dollars per bin, is [tex] p(x) = 425 - 2x [/tex].

1. Find the cost function.
[tex]\[ C(x) = \][/tex]

2. Find the revenue function.
[tex]\[ R(x) = \][/tex]

3. Find the profit function.
[tex]\[ P(x) = \][/tex]

4. At what quantity is the smallest break-even point?
- Select an answer.

Sagot :

GinnyAnswer
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]
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