The pricing policy of a company follows the demand equation p=D(x). D(x) being the price per unit when x units are demanded. After studying the market trends the company determines the price function that is given by D(x)= 2000-4x. If the product is to be marketed the company will incur a fixed cost of $60000 and will have to pay $600 for each unit that is produced. At what sales level the company expect to recover its cost

Good

Well, let's see if we can find out when this company will be laughing all the way to the bank!

To determine when the company will recover its costs, we need to find the sales level, x, at which the revenue will cover all the expenses.

The revenue is the product of the price per unit, D(x), and the number of units sold, x. So, the revenue function, R(x), can be expressed as R(x) = x * D(x).

The total cost, C(x), is the sum of the fixed cost and the variable cost per unit multiplied by the number of units, which is $60,000 + ($600 * x).

To find when the company recovers its costs, we need to equate the revenue and the total cost:

x * D(x) = $60,000 + $600 * x

Let's substitute D(x) with its given equation:

x * (2000-4x) = $60,000 + $600 * x

2000x - 4x^2 = $60,000 + $600 * x

Now, let's rearrange the equation to solve for x:

4x^2 + (600 - 2000)x + $60,000 = 0

Using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), where a = 4, b = (600 - 2000), and c = $60,000:

x = [-(600 - 2000) ± √((600 - 2000)^2 - 4 * 4 * $60,000)] / (2 * 4)

After a few calculations, we find that the solutions for x are x ≈ 335.8 and x ≈ 352.2.

Now, since we can't sell a fraction of a unit, the company expects to recover its costs at a sales level of either 335 units or 352 units.

I hope this helps the company keep track of its finances and have a good laugh along the way!

To determine the sales level at which the company expects to recover its cost, we need to find the break-even point where the total cost equals the total revenue.

Let's break down the total cost and total revenue components:

Total Cost (TC) = Fixed Cost (FC) + Variable Cost per unit (VC) * Number of units sold (x)

TC = FC + VC * x

Given:
Fixed Cost (FC) = $60,000
Variable Cost per unit (VC) = $600

Total Revenue (TR) = Price per unit (p) * Number of units sold (x)

TR = D(x) * x

Demand Equation:
D(x) = 2000 - 4x

Now, we'll set TC equal to TR and solve for x:

FC + VC * x = D(x) * x

Substituting FC and VC values:

$60,000 + $600 * x = (2000 - 4x) * x

Let's simplify this equation:

$60,000 + $600x = 2000x - 4x^2

Rearranging the equation:

4x^2 - 2,000x + 600x - $60,000 = 0

Combining like terms:

4x^2 - 1,400x - $60,000 = 0

Now, we can solve this quadratic equation for x using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Here, a = 4, b = -1,400, and c = -60,000

x = (-(-1,400) ± √((-1,400)^2 - 4 * 4 * -60,000)) / (2 * 4)

x = (1,400 ± √(1,960,000 - (-960,000))) / 8

x = (1,400 ± √2,920,000) / 8

x = (1,400 ± 1,710.4) / 8

We have two possible solutions:

x1 = (1,400 + 1,710.4) / 8 ≈ 321.3 (rounded to the nearest whole number)
x2 = (1,400 - 1,710.4) / 8 ≈ -190.4 (rounded to the nearest whole number)

Since the number of units sold cannot be negative, the company expects to recover its cost at approximately 321 units sold.

To determine at what sales level the company expects to recover its cost, we need to find the value of x when the revenue equals the total cost.

The revenue is the product of the price per unit (D(x)) and the number of units sold (x). The total cost is the sum of the fixed cost and the variable cost per unit multiplied by the number of units sold.

Given:
Price function: D(x) = 2000 - 4x
Fixed cost: $60,000
Variable cost per unit: $600

Let's consider the revenue equation:
Revenue = Price per unit * Number of units sold = D(x) * x

Total Cost = Fixed Cost + Variable Cost per unit * Number of units sold = $60,000 + $600 * x

To find the sales level where the company expects to recover its cost, we'll set the revenue equal to the total cost and solve for x:

D(x) * x = $60,000 + $600 * x

Substitute the price function D(x) = 2000 - 4x:

(2000 - 4x) * x = $60,000 + $600 * x

Expand and simplify:

2000x - 4x^2 = $60,000 + $600x

Rearrange the equation:

4x^2 + 600x - 2000x - $60,000 = 0

Combine like terms:

4x^2 - 1400x - $60,000 = 0

Now, we have a quadratic equation. To solve for x, we can use factoring, completing the square, or the quadratic formula. Once we determine the roots, we'll take the positive value since x represents the number of units sold.

Once we calculate the value of x, we'll have the sales level at which the company expects to recover its cost.