Answered

IDNLearn.com makes it easy to find precise answers to your specific questions. Ask anything and receive thorough, reliable answers from our community of experienced professionals.

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 :

GinnyAnsweridn
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]
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Your questions find answers at IDNLearn.com. Thanks for visiting, and come back for more accurate and reliable solutions.