A commodity has a demand function modeled by 
p = 112 − 0.5x
 and a total cost function modeled by 
C = 50x + 37.75,
 where x is the number of units.
(a)
What unit price (in dollars) yields a maximum profit?
$ 
81
 
Correct: Your answer is correct.
 per unit
(b)
When the profit is maximized, what is the average cost (in dollars) per unit? (Round your answer to two decimal places.)


To solve part (b), we first need to recall the formula for profit. Profit is given by:

\[
\text{Profit} = \text{Revenue} - \text{Cost}
\]

### Step 1: Calculate the Revenue Function
Revenue is the price per unit times the number of units sold. From the demand function, the price $ p $ is:

\[
p = 112 - 0.5x
\]

Thus, the revenue $ R(x) $ is:

\[
R(x) = p \cdot x = (112 - 0.5x) \cdot x
\]

Expanding this:

\[
R(x) = 112x - 0.5x^2
\]

### Step 2: Write the Profit Function
The total cost function is:

\[
C(x) = 50x + 37.75
\]

Thus, the profit function $ P(x) $ is:

\[
P(x) = R(x) - C(x)
\]

Substitute the expressions for $ R(x) $ and $ C(x) $:

\[
P(x) = (112x - 0.5x^2) - (50x + 37.75)
\]

Simplify:

\[
P(x) = 112x - 0.5x^2 - 50x - 37.75
\]

\[
P(x) = 62x - 0.5x^2 - 37.75
\]

### Step 3: Maximize the Profit
To maximize profit, we take the derivative of the profit function and set it equal to zero.

\[
\frac{dP(x)}{dx} = 62 - x
\]

Set the derivative equal to zero to find the critical point:

\[
62 - x = 0
\]

\[
x = 62
\]

### Step 4: Find the Unit Price When Profit is Maximized
Now that we know $ x = 62 $ maximizes the profit, we substitute $ x = 62 $ into the demand function to find the price:

\[
p = 112 - 0.5(62) = 112 - 31 = 81
\]

Thus, the unit price that maximizes profit is **$81**.

### Step 5: Calculate the Average Cost When Profit is Maximized
The average cost is given by:

\[
\text{Average Cost} = \frac{C(x)}{x}
\]

Substitute $ x = 62 $ into the cost function:

\[
C(62) = 50(62) + 37.75 = 3100 + 37.75 = 3137.75
\]

Now calculate the average cost:

\[
\text{Average Cost} = \frac{3137.75}{62} \approx 50.6
\]

### Final Answer:
The average cost per unit when the profit is maximized is **$50.60**.

---

Would you like more details or have any further questions on this? Here are a few related questions:

1. How do you interpret the profit maximization in terms of the demand and cost functions?
2. What would happen to the unit price if the total cost function had a higher fixed cost?
3. How does increasing the number of units affect the average cost?
4. How can you determine the price elasticity of demand from the demand function?
5. What would be the effect of reducing the cost per unit on the profit maximization?

**Tip:** Always ensure that you're maximizing profit by taking the derivative of the profit function and solving for critical points.

A commodity has a demand function modeled by p = 112 − 0.5x and a total cost function modeled by C = 50x + 37.75, where x is the number of units. (a) What unit price (in dollars) yields a maximum profit? (b) When the profit is maximized, what is the average cost (in dollars) per unit?

Learn how to calculate the unit price that maximizes profit and find the average cost per unit when profit is maximized for a commodity with a given demand and total cost function. The solution uses derivatives to find the optimal price and cost values. Suitable for students in Grades 11-12 studying optimization and economics.

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