The total cost, in millions of dollars, of producing x thousand units of an item is C(x) = 4(x − 1)^2 + 4.

The revenue (in millions of dollars) from selling x thousand units of the item is R(x) = 10x. What does this tell you about the price of each unit?

Give your answer without commas.

Would I use R(X) as the input for c(x) so it would be c(x) = 4 (10x -1)^2 + ?

when will I lose money

0 < x < 1/2 and x > 4

or

1/2 < x < 4

Whether to calculate the price is correct and when will I be losing money

X value to make profit would be

1/2 < x <4

Break even point

x = 1/2 and x = 4

If this part of my work helps at all. Just looking at this point for the price of each unit and the check portion for when I will lose money

The price of each unit can be determined by comparing the revenue and cost functions. In this case, the revenue function R(x) = 10x represents the total revenue in millions of dollars from selling x thousand units of the item.

To find the cost per unit, we can divide the total cost C(x) by the number of units x:
Cost per unit = C(x) / x

Substituting the given cost function C(x) = 4(x − 1)^2 + 4, we have:
Cost per unit = [4(x − 1)^2 + 4] / x

To determine when you will lose money, you need to find the range of values for x where the cost is greater than the revenue (C(x) > R(x)).

Comparing the cost and revenue functions, we have:
4(x − 1)^2 + 4 > 10x

Simplifying the inequality, we get:
4x^2 - 18x + 20 > 0

Factoring the quadratic, we have:
(x - 1)(4x - 20) > 0

This inequality holds true when either both factors are positive or both factors are negative.

Case 1: (x - 1) > 0 and (4x - 20) > 0
This implies x > 1 and x > 5. However, since x > 5 contradicts x > 1, we only consider the range x > 1.

Case 2: (x - 1) < 0 and (4x - 20) < 0
This implies x < 1 and x < 5. However, since x < 1 contradicts x < 5, we only consider the range x < 5.

Therefore, the range of x-values when you will lose money is 1 < x < 5.
In interval notation, this can be expressed as (1, 5).

To determine the price of each unit, we need to compare the revenue and the total cost functions.

From the given information:
- The revenue function is R(x) = 10x million dollars.
- The total cost function is C(x) = 4(x - 1)^2 + 4 million dollars.

The revenue from selling x thousand units is given by R(x), which represents the income earned. The cost of producing x thousand units is given by C(x), which represents the expenses incurred in producing the items.

To find the price of each unit, we need to calculate the difference between revenue and cost, divided by the number of units sold.

Price per unit (P) = (Revenue - Total cost)/Number of units

Substituting the functions R(x) and C(x) into the equation, we get:

P = (10x - 4(x - 1)^2 - 4)/x

Simplifying this expression is the next step. However, it's important to note that the provided information does not explicitly indicate a specific value for x. Therefore, we cannot determine a numerical answer for the price per unit without knowing the value of x.

Moving on to the second part of your question, to determine when you will lose money, you need to find the range of x values for which the total cost C(x) is greater than the revenue R(x).

To find this range, compare the two functions:

10x < 4(x - 1)^2 + 4

Rearranging the equation:

0 < 4(x - 1)^2 + 4 - 10x

Next, simplify and solve for x. It's possible to use algebraic methods or graphing techniques to find the range of x values for which the cost is greater than the revenue.

After solving the inequality, the range of x values when you will lose money is:
0 < x < 1/2 and x > 4
or
1/2 < x < 4

In these ranges, the total cost will be greater than the revenue, resulting in a loss.