https://file.aiquickdraw.com/mathbot/file/17267646699222457hm32.jpg

Let's solve the questions shown in the image step by step.

### Given:
1. **Cost function**: \( C(x) = -x^3 + 8x^2 - 4x - 5 \)
2. **Revenue function**: \( R(x) = -x^3 + 7x^2 \)
3. **Demand function**: \( D(x) = -x^2 + 7x \)
4. **Supply function**: \( S(x) = x^2 - x \)

Where \( x \) represents the number of units sold, and \( 0 < x \leq 15 \).

### (a) What is the profit if 6 units are sold?
The profit \( P(x) \) is given by the formula:
\[
P(x) = R(x) - C(x)
\]
We will substitute \( x = 6 \) into both the revenue and cost functions and calculate the profit.

### (b) How many units must be sold for the company to break even? Does the company make a profit if 6 units are sold?
To break even, profit should be zero, i.e., \( P(x) = 0 \). We will solve for \( x \) by equating \( R(x) = C(x) \).

### (c) Find the market equilibrium point: find both the number of units being sold at the market equilibrium point and the price per unit.
The market equilibrium occurs when demand equals supply, i.e., \( D(x) = S(x) \). We will solve for \( x \) to find the number of units at equilibrium and substitute that value into the demand (or supply) function to find the price per unit.

Let me begin with part (a) and calculate the profit if 6 units are sold.The profit when 6 units are sold is \(-7\) dollars, indicating a loss of 7 dollars.

Let's now move on to part (b) and determine how many units must be sold for the company to break even. This happens when \( P(x) = 0 \), meaning \( R(x) = C(x) \). I'll solve for \( x \).The break-even points occur when \( x = -1 \) and \( x = 5 \). Since \( x \) must be positive (number of units sold), the relevant break-even point is \( x = 5 \).

Thus, the company breaks even when 5 units are sold.

Now, I'll check if the company makes a profit when 6 units are sold (as calculated earlier). Since the profit at 6 units is \(-7\) dollars, the company does **not** make a profit when 6 units are sold.

Finally, let's solve part (c) to find the market equilibrium by setting \( D(x) = S(x) \).The market equilibrium occurs when \( x = 0 \) or \( x = 4 \). Since \( x = 0 \) is not a practical solution (no units sold), the market equilibrium point is when 4 units are sold.

Now, let's find the price per unit at equilibrium by substituting \( x = 4 \) into either the demand or supply function (both will give the same result). I'll compute the price next.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can substitute \( x = 4 \) into either the demand function \( D(x) = -x^2 + 7x \) or the supply function \( S(x) = x^2 - x \) to find the price at equilibrium. Here's how you can calculate it:

1. Demand function at equilibrium:
   \[
   D(4) = -(4)^2 + 7(4) = -16 + 28 = 12
   \]

Thus, the price per unit at equilibrium is 12 dollars.

Let me know if you'd like any further clarification or details!

Here are 5 follow-up questions to expand on this topic:
1. How does a company typically find the optimal number of units to produce for maximum profit?
2. What factors affect the supply and demand functions in real-world applications?
3. What would happen to the break-even point if fixed costs (e.g., the constant in the cost function) were reduced?
4. How does the equilibrium price change if the supply curve shifts due to an external factor?
5. How would you calculate the marginal cost and marginal revenue from the given functions?

**Tip:** Always recheck whether all break-even points are valid within the practical domain of your problem (e.g., non-negative units).

Suppose the following functions describe the operations of a company. Assume that x is the number of units of a certain product and that 0 < x <= 15.
Cost (in dollars): C(x) = -x^3 + 8x^2 - 4x - 5
Revenue (in dollars): R(x) = -x^3 + 7x^2
Demand (in dollars per unit): D(x) = -x^2 + 7x
Supply (in dollars per unit): S(x) = x^2 - x
(a) What is the profit if 6 units are sold?
(b) How many units must be sold for the company to break even? Does the company make a profit if 6 units are sold?
(c) Find the market equilibrium point: find both the number of units being sold at the market equilibrium point and the price per unit.

Solve the problem involving cost, revenue, demand, and supply functions to find profit, break-even point, and market equilibrium. This exercise covers key algebraic concepts for high school or early college economics courses, including profit calculation and equilibrium analysis.

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