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The given demand equation for the price $ p $ and the quantity $ x $ sold is:

\[
p = -11x + 187
\]

Let's break down and solve each part of the question.

### Part (a)
**How much should be charged for the television set if there are 80 television sets in stock?**

We substitute $ x = 80 $ into the demand equation to find the price $ p $.

\[
p = -11(80) + 187
\]
\[
p = -880 + 187
\]
\[
p = -693
\]

So, the price should be **$-693**, which indicates an unrealistic negative value. This likely reflects that 80 units is beyond the scope of realistic stock quantities, as a price less than or equal to zero isn't feasible in real-world scenarios.

### Part (b)
**What quantity $ x $ will maximize revenue? What is the maximum revenue?**

Revenue $ R(x) $ is the product of price and quantity, so:

\[
R(x) = p(x) \cdot x = (-11x + 187)x
\]
\[
R(x) = -11x^2 + 187x
\]

This is a quadratic function, which opens downward (since the coefficient of $ x^2 $ is negative), meaning it has a maximum. To find the value of $ x $ that maximizes revenue, we use the formula for the vertex of a parabola:

\[
x = \frac{-b}{2a}
\]

Here, $ a = -11 $ and $ b = 187 $, so:

\[
x = \frac{-187}{2(-11)} = \frac{187}{22} \approx 8.5
\]

So, approximately **8.5** television sets will maximize revenue.

To find the maximum revenue, substitute $ x = 8.5 $ back into the revenue equation:

\[
R(8.5) = -11(8.5)^2 + 187(8.5)
\]
\[
R(8.5) = -11(72.25) + 187(8.5)
\]
\[
R(8.5) = -794.75 + 1589.5 = 794.75
\]

The maximum revenue is **$794.75**.

### Part (c)
**What price should be charged in order to maximize revenue?**

To find the price that corresponds to the maximum revenue, substitute $ x = 8.5 $ into the demand equation:

\[
p = -11(8.5) + 187
\]
\[
p = -93.5 + 187 = 93.5
\]

So, the price to maximize revenue is **$93.50**.

---

Let me know if you'd like more details or have any questions.

Here are 5 follow-up questions to expand on the topic:
1. What is the significance of the negative price in part (a) for large quantities?
2. How does the vertex formula help in determining maximum revenue?
3. What happens to the revenue function if the cost per unit is added?
4. How would the demand equation change if external factors influence the market?
5. Can the model account for production costs, and how would that affect the revenue?

**Tip:** For quadratic functions, the vertex provides key insights into maxima or minima, depending on the direction of the parabola.

The price p and the quantity x sold of a small flat-screen television set obeys the demand equation p = -11x + 187. How much should be charged if there are 80 television sets in stock? What quantity maximizes revenue, and what is the maximum revenue? What price should be charged to maximize revenue?

Learn how to calculate the price for a small flat-screen TV set using the demand equation p = -11x + 187. This problem explains how to find the quantity that maximizes revenue and determine the corresponding maximum revenue. Ideal for high school students studying algebra.

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