The manager of a furniture factory finds that it costs $ 2300 to manufacture 110 chairs in one day and $ 5000 to produce 295 chairs in one day.

This problem can be tackled using a linear cost model, where the cost to manufacture chairs is a linear function of the number of chairs produced. The general form of a linear equation is:

\[
C(x) = mx + b
\]

Where:
- $ C(x) $ is the total cost to produce $ x $ chairs.
- $ m $ is the marginal cost (cost to produce each additional chair).
- $ b $ is the fixed cost (cost when no chairs are produced).

### Step 1: Set up two equations based on the given information.
We are given two points:
1. $ C(110) = 2300 $ (It costs $2300 to produce 110 chairs).
2. $ C(295) = 5000 $ (It costs $5000 to produce 295 chairs).

These two points will allow us to solve for the values of $ m $ and $ b $.

### Step 2: Set up the system of equations.
From the given points:

\[
2300 = 110m + b
\]
\[
5000 = 295m + b
\]

### Step 3: Solve the system of equations.
We can solve this system to find $ m $ and $ b $ by eliminating one of the variables. Let's subtract the first equation from the second:

\[
(5000 - 2300) = (295m + b) - (110m + b)
\]
\[
2700 = 185m
\]
\[
m = \frac{2700}{185} = 14.59
\]

Thus, the marginal cost to produce each additional chair is approximately $14.59.

### Step 4: Solve for the fixed cost $ b $.
Now, substitute $ m = 14.59 $ back into one of the original equations. Let's use $ 2300 = 110m + b $:

\[
2300 = 110(14.59) + b
\]
\[
2300 = 1604.9 + b
\]
\[
b = 2300 - 1604.9 = 695.1
\]

The fixed cost $ b $ is approximately $695.10.

### Final equation:
The cost function is:

\[
C(x) = 14.59x + 695.10
\]

This means the total cost to manufacture $ x $ chairs is approximately $14.59 per chair, plus a fixed cost of $695.10.

Would you like more details on any part of this solution? Here are five questions to deepen your understanding:

1. How can you interpret the marginal cost and fixed cost in practical terms?
2. What would be the cost of producing 200 chairs?
3. Can you derive the number of chairs that can be produced for a given budget?
4. How does the cost model change if the marginal cost decreases as production increases?
5. What other factors might influence production costs in real-world scenarios?

**Tip:** When solving systems of equations, substitution or elimination methods can be useful for finding unknown variables efficiently.

The manager of a furniture factory finds that it costs $2300 to manufacture 110 chairs in one day and $5000 to produce 295 chairs in one day.

Learn how to determine the linear cost function for manufacturing chairs at a furniture factory. This problem demonstrates how to solve systems of linear equations using real-world data. $2300 for 110 chairs and $5000 for 295 chairs. Suitable for students in Grades 9-12.

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