A piano manufacturer has a daily fixed cost of $1,600 and a marginal cost of $1,800 per piano. Find the cost 

*C*(*x*)

 of manufacturing *x* pianos in one day.

*C*(*x*) =  
 
 
Use your function to answer the following questions.

(a)

On a given day, what is the cost of manufacturing three pianos?

$  

(b)

What is the cost of manufacturing the third piano that day?

$ 

(c)

What is the cost of manufacturing the 11th piano that day?

$ 

(d)

What is the variable cost?

$  
 
What is the fixed cost?

$ 

What is the marginal cost?

$  per piano

Let's solve the given problem step-by-step.

### Given Information:
1. **Fixed cost**: $1,600 per day.
2. **Marginal cost**: $1,800 per piano.

### Step 1: Formulate the Cost Function
The cost function $ C(x) $ represents the total cost of manufacturing $ x $ pianos in one day. This includes both the fixed cost and the variable costs, which depend on the number of pianos manufactured.

The cost function can be expressed as:
\[
C(x) = \text{Fixed Cost} + (\text{Marginal Cost} \times x)
\]

Plugging in the given values:
\[
C(x) = 1600 + 1800x
\]

### Step 2: Use the Function to Answer the Questions

#### (a) Cost of Manufacturing Three Pianos
To find the cost of manufacturing three pianos, substitute $ x = 3 $ into the cost function:

\[
C(3) = 1600 + 1800 \times 3
\]

Calculating this:

\[
C(3) = 1600 + 5400 = 7000
\]

So, the cost of manufacturing three pianos in one day is **$7,000**.

#### (b) Cost of Manufacturing the Third Piano
The cost of manufacturing the third piano is the marginal cost since it represents the additional cost incurred when one more piano is produced. The marginal cost is given as:

\[
\text{Marginal Cost} = 1800
\]

So, the cost of manufacturing the third piano that day is **$1,800**.

#### (c) Cost of Manufacturing the 11th Piano
Similarly, the cost of manufacturing the 11th piano is also the marginal cost, which is:

\[
\text{Marginal Cost} = 1800
\]

So, the cost of manufacturing the 11th piano that day is **$1,800**.

#### (d) Variable, Fixed, and Marginal Costs
- **Variable Cost**: The total variable cost depends on the number of pianos manufactured, which is:

\[
\text{Variable Cost} = 1800x
\]

- **Fixed Cost**: This is the cost that does not change regardless of the number of pianos produced:

\[
\text{Fixed Cost} = 1600
\]

- **Marginal Cost**: The cost of producing one additional piano is:

\[
\text{Marginal Cost} = 1800 \text{ per piano}
\]

### Summary of Answers:

1. $ C(x) = 1600 + 1800x $

2. **(a)** The cost of manufacturing three pianos: **$7,000**  
3. **(b)** The cost of manufacturing the third piano: **$1,800**  
4. **(c)** The cost of manufacturing the 11th piano: **$1,800**  
5. **(d)** Variable Cost: **$1800x**  
   Fixed Cost: **$1600**  
   Marginal Cost: **$1,800 per piano**  

Would you like more details, or do you have any other questions?

### Related Questions:
1. What would be the cost of manufacturing 10 pianos?
2. How many pianos must be manufactured to have a total cost of $10,000?
3. What is the average cost per piano if 5 pianos are produced?
4. How does the cost function change if the fixed cost increases to $2,000?
5. What is the breakeven point if each piano is sold for $2,500?

**Tip:** Remember, the marginal cost helps you determine how much each additional unit will cost to produce!

A piano manufacturer has a daily fixed cost of $1,600 and a marginal cost of $1,800 per piano. Find the cost C(x) of manufacturing x pianos in one day. Use your function to answer the following questions: (a) On a given day, what is the cost of manufacturing three pianos? (b) What is the cost of manufacturing the third piano that day? (c) What is the cost of manufacturing the 11th piano that day? (d) What is the variable cost? What is the fixed cost? What is the marginal cost?

Learn how to calculate the total cost of manufacturing pianos with fixed and marginal costs. This detailed guide includes solving the cost function C(x) = 1600 + 1800x to find the cost of manufacturing specific pianos, such as the third and 11th pianos. Suitable for students in Grades 9-12.

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