A company produces a special new type of TV.  The company has fixed costs of ​$498 comma 000498,000​, and it costs ​$12001200 to produce each TV.  The company projects that if it charges a price of ​$26002600 for the​ TV, it will be able to sell 800800 TVs.  If the company wants to sell 850850 ​TVs, however, it must lower the price to ​$23002300.  Assume a linear demand.
How many TVs must the company sell to earn ​$2 comma 280 comma 0002,280,000 in​ revenue?
It needs to sell enter your response here TVs.
​(Round answer to nearest integer.  If more than one​ answer, separate with a​ comma.)

To solve this problem, we need to use the following steps:

### Step 1: Set up the demand function.
Assume a linear demand function:
\[
p(x) = mx + b
\]
where $ p(x) $ is the price of each TV when $ x $ TVs are sold, $ m $ is the slope, and $ b $ is the y-intercept.

We are given two points to define this line:
- When 800 TVs are sold, the price is $2600.
- When 850 TVs are sold, the price is $2300.

Using these two points $(800, 2600)$ and $(850, 2300)$, we can first calculate the slope $ m $:

\[
m = \frac{2300 - 2600}{850 - 800} = \frac{-300}{50} = -6
\]

Thus, the demand function so far is:
\[
p(x) = -6x + b
\]

### Step 2: Find the intercept $ b $.
Using the point $(800, 2600)$ to solve for $ b $:

\[
2600 = -6(800) + b
\]
\[
2600 = -4800 + b
\]
\[
b = 2600 + 4800 = 7400
\]

So the demand function is:
\[
p(x) = -6x + 7400
\]

### Step 3: Write the revenue function.
Revenue $ R(x) $ is given by the number of TVs sold $ x $ times the price $ p(x) $:

\[
R(x) = x \cdot p(x) = x(-6x + 7400) = -6x^2 + 7400x
\]

### Step 4: Solve for the number of TVs to sell to earn $2,280,000 in revenue.
We are given that the revenue should be $2,280,000, so:

\[
-6x^2 + 7400x = 2,280,000
\]

Rearranging the equation:
\[
-6x^2 + 7400x - 2,280,000 = 0
\]

### Step 5: Solve the quadratic equation.
We will now solve the quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where $ a = -6 $, $ b = 7400 $, and $ c = -2,280,000 $.

Substitute the values into the quadratic formula:
\[
x = \frac{-7400 \pm \sqrt{7400^2 - 4(-6)(-2,280,000)}}{2(-6)}
\]
Now, calculate the discriminant:
\[
7400^2 = 54760000
\]
\[
4(-6)(-2,280,000) = 54720000
\]
\[
\text{Discriminant} = 54760000 - 54720000 = 40000
\]
Now solve for $ x $:
\[
x = \frac{-7400 \pm \sqrt{40000}}{-12}
\]
\[
x = \frac{-7400 \pm 200}{-12}
\]

This gives two possible solutions:
\[
x_1 = \frac{-7400 + 200}{-12} = \frac{-7200}{-12} = 600
\]
\[
x_2 = \frac{-7400 - 200}{-12} = \frac{-7600}{-12} = 633.33
\]

Since $ x $ must be an integer, the solutions are $ x = 600 $ and $ x = 633 $.

### Conclusion:
The company must sell **600 or 633 TVs** to earn $2,280,000 in revenue.

Would you like more details on any part of the process? Here are 5 related questions:
1. How is the slope of the demand function derived from two price points?
2. What does the y-intercept represent in this context?
3. Can you explain how the revenue function was constructed?
4. Why does the discriminant help us determine the number of solutions?
5. What are the real-world implications of a quadratic revenue function?

**Tip**: When solving quadratic equations, always check if rounding is necessary, especially in business or economic contexts where partial units may not be practical.

A company produces a special new type of TV. The company has fixed costs of $498,000, and it costs $1200 to produce each TV. The company projects that if it charges a price of $2600 for the TV, it will be able to sell 800 TVs. If the company wants to sell 850 TVs, however, it must lower the price to $2300. Assume a linear demand. How many TVs must the company sell to earn $2,280,000 in revenue?

This problem explores how to calculate the number of TVs a company needs to sell to earn $2,280,000 in revenue. Using a linear demand function and solving a quadratic equation, we determine that the company must sell either 600 or 633 TVs. Suitable for students in Grades 9-12.

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