A company determines that its demand function is given by p left parenthesis q right parenthesis equals negative 0.5 q plus 1074 and that its cost function is given by Upper C left parenthesis q right parenthesis equals 109.9 q plus 22246.  What profit can the company earn by producing and selling 724 ​items?

To determine the company's profit by producing and selling 724 items, we need to calculate the revenue and cost, then subtract the cost from the revenue.

### Step 1: Revenue Function
The revenue $ R(q) $ is the product of the demand function $ p(q) $ and the quantity $ q $.

- Demand function: $ p(q) = -0.5q + 1074 $
- Revenue function: $ R(q) = p(q) \times q = (-0.5q + 1074)q $

Now, we calculate the revenue when $ q = 724 $:

\[
R(724) = (-0.5 \times 724 + 1074) \times 724
\]

### Step 2: Cost Function
The cost function is given by $ C(q) = 109.9q + 22246 $.

We calculate the cost when $ q = 724 $:

\[
C(724) = 109.9 \times 724 + 22246
\]

### Step 3: Profit Function
The profit function $ \pi(q) $ is the difference between revenue and cost:

\[
\pi(q) = R(q) - C(q)
\]

We will now calculate the profit for $ q = 724 $.

Let's compute the values.The company's profit from producing and selling 724 items is \$413,674.40.

Here is a breakdown of the calculation:
- Revenue from selling 724 items: \$515,488.00
- Cost of producing 724 items: \$101,813.60
- Profit: \$515,488.00 - \$101,813.60 = \$413,674.40

Would you like more details or further clarification?

Here are 5 related questions to consider:
1. How does the profit change if the number of items produced changes?
2. What happens to profit if the cost per item increases?
3. How would a change in demand affect the revenue function?
4. What is the break-even point where profit equals zero?
5. How does a price reduction strategy impact overall profit?

**Tip:** When analyzing profit, always consider both marginal revenue and marginal cost to optimize production levels.

A company determines that its demand function is given by p(q) = -0.5q + 1074 and that its cost function is given by C(q) = 109.9q + 22246. What profit can the company earn by producing and selling 724 items?

Calculate the profit for a company producing and selling 724 items using a demand function of p(q) = -0.5q + 1074 and a cost function of C(q) = 109.9q + 22246. Learn how to determine revenue, cost, and profit with a detailed solution. Ideal for students in Grades 10-12.

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