A company produces a special new type of TV.  The company has fixed costs of ​$480 comma 000​, and it costs ​$1400 to produce each TV.  The company projects that if it charges a price of ​$2600 for the​ TV, it will be able to sell 850 TVs.  If the company wants to sell 900 ​TVs, however, it must 
lower the price to ​$2300.  Assume a linear demand.
What price should the company charge to earn a profit of ​$1 comma 080 comma 000​?

We are given the following information about the company producing the new type of TV:

- Fixed costs: \$480,000
- Variable cost per TV: \$1,400
- Selling price and demand relationship:
  - At a price of \$2,600, the company can sell 850 TVs.
  - At a price of \$2,300, the company can sell 900 TVs.

To determine the price at which the company should charge to earn a profit of \$1,080,000, we can break the problem into steps:

### Step 1: Find the Demand Equation

Since we assume the demand is linear, we can express the demand (quantity sold) as a function of price using the equation for a line: 

\[
q = mp + b
\]

Where:
- $q$ is the quantity of TVs sold,
- $p$ is the price charged,
- $m$ is the slope of the demand line,
- $b$ is the intercept (when $p = 0$).

We can use the two data points given: (2600, 850) and (2300, 900). The slope $m$ of the line is calculated as:

\[
m = \frac{900 - 850}{2300 - 2600} = \frac{50}{-300} = -\frac{1}{6}
\]

Thus, the demand equation is:

\[
q = -\frac{1}{6}p + b
\]

### Step 2: Solve for $b$

We can now use one of the given points (let's use $p = 2600$, $q = 850$) to solve for $b$:

\[
850 = -\frac{1}{6}(2600) + b
\]

\[
850 = -\frac{2600}{6} + b
\]

\[
850 = -433.33 + b
\]

\[
b = 850 + 433.33 = 1283.33
\]

Thus, the demand equation is:

\[
q = -\frac{1}{6}p + 1283.33
\]

### Step 3: Find the Number of TVs Sold for the Desired Profit

The company wants to earn a profit of \$1,080,000. To find the number of TVs $q$ that need to be sold to achieve this profit, we need to first write an expression for the profit.

The revenue $R$ is given by:

\[
R = p \times q
\]

The total cost $C$ is:

\[
C = \text{Fixed cost} + (\text{Variable cost per TV} \times q)
\]
\[
C = 480000 + 1400q
\]

The profit is:

\[
\text{Profit} = R - C
\]

Substituting for $R$ and $C$:

\[
1,080,000 = p \times q - (480000 + 1400q)
\]

\[
1,080,000 = pq - 480000 - 1400q
\]

Since we have the demand equation $q = -\frac{1}{6}p + 1283.33$, we can substitute this into the profit equation and solve for the price $p$.

Let’s solve this equation next.The company should charge approximately **\$3,800** to achieve a profit of \$1,080,000. 

Would you like more details on the solution, or do you have any questions? 

Here are five related questions that build on this problem:
1. How would the price change if the company aimed for a profit of \$2,000,000?
2. What would be the break-even price for this product?
3. How does the fixed cost impact the price required to achieve a target profit?
4. How would increasing the variable cost per TV affect the price required to sell?
5. How does the linear demand assumption affect the model's accuracy in the real world?

**Tip:** A higher price may decrease demand, so it's essential to balance pricing strategy with the projected demand in such models.

A company produces a special new type of TV. The company has fixed costs of $480,000, and it costs $1,400 to produce each TV. The company projects that if it charges a price of $2,600 for the TV, it will be able to sell 850 TVs. If the company wants to sell 900 TVs, however, it must lower the price to $2,300. Assume a linear demand. What price should the company charge to earn a profit of $1,080,000?

Learn how to calculate the price a company should charge to earn a specific profit by analyzing the linear demand for a new type of TV. This problem covers fixed and variable costs, profit calculations, and linear equations. Suitable for students in Grades 10-12.

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