The weekly cost of manufacturing x telephones per week is found by a manufacturer to be C(x) = 500 + 20x = x^2 dollars. the telephone can be sold at a price p = $80 each. find the manufacturing break-even production level and for what production levels will the manufacturer experience a profit

telephones are not cheap to make, young whoppa

breakeven occurs when cost = revenue

don't know whether you have + or minus x^2. On my keyboard, + and = are the same key, so I'll assume "+".

x^2 + 20x + 500 = 80x
x^2 - 60x + 500 = 0
(x-10)(x-50) = 0

So, for x between 10 and 50, cost is less than revenue.

This is a strange model. Usually as quantity increases, cost goes down.

Maybe you should have had C(x) = 500 + 20x - 1/x^2
or something. That would mean there's a fixed cost of $500 just for making phones, and a $20/phone cost for materials, say, and a decreasing manufacturing cost as quantity goes up. (efficiency of scale)

In that case, we'd have

500 + 20x - 1/(x^2+1) = 80x

That shows costs greater than revenue until x = 8.3 or so, then revenue is greater than costs.

To find the manufacturing break-even production level, we need to determine the production level (x) at which the cost (C(x)) is equal to the revenue obtained from selling that number of telephones.

The revenue is given by the equation:
R(x) = p * x
where p is the price of each telephone, given as $80.

So, the revenue equation becomes:
R(x) = 80x

The break-even point occurs when the cost is equal to the revenue. Therefore, we can set up the equation:
C(x) = R(x)

Given that C(x) = 500 + 20x + x^2 and R(x) = 80x, we can substitute these values to find the break-even point.

500 + 20x + x^2 = 80x

Rearrange the equation:
x^2 + 20x + 500 = 80x

Now, we can solve the quadratic equation by bringing everything to one side:

x^2 + 20x - 80x + 500 = 0

Combine like terms:
x^2 - 60x + 500 = 0

Next, we can factor or utilize the quadratic formula to solve this equation. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

For our equation, a = 1, b = -60, and c = 500. Substituting these values into the formula, we have:

x = (-(-60) ± √((-60)^2 - 4(1)(500))) / (2(1))

x = (60 ± √(3600 - 2000)) / 2

x = (60 ± √(1600)) / 2

x = (60 ± 40) / 2

This gives us two potential solutions:

x₁ = (60 + 40) / 2 = 100 / 2 = 50
x₂ = (60 - 40) / 2 = 20 / 2 = 10

Since we are looking for the manufacturing break-even point, we choose the positive value for x, which is x = 50. Therefore, the manufacturing break-even production level is 50 telephones per week.

To determine the production levels at which the manufacturer will experience a profit, we need to compare the revenue (R(x)) to the cost (C(x)).

For profit to occur, the revenue must be greater than the cost, i.e., R(x) > C(x).

Given R(x) = 80x and C(x) = 500 + 20x + x^2, we can set up the inequality:

80x > 500 + 20x + x^2

Simplifying:
x^2 + 60x - 500 > 0

Similar to earlier, we can solve this quadratic inequality using factoring or the quadratic formula. By solving, we find that the production levels which yield a profit are:

x < -50 or x > 10

This means that the manufacturer will experience a profit when the production level (x) is less than -50 or greater than 10.