A piano manufacturer has a daily fixed cost of $1,300 and a marginal cost of $1,600 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

(e)

Graph *C* as a function of *x*. (Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.)

Let's go through this step by step:

### 1. **Cost Function $C(x)$ for Manufacturing $x$ Pianos in One Day**

- **Fixed cost**: This is the cost the manufacturer incurs regardless of the number of pianos produced. In this case, the fixed cost is $1,300.
- **Marginal cost**: This is the cost of producing one additional piano, which is $1,600 per piano.

The total cost function $ C(x) $, where $ x $ is the number of pianos produced, can be written as:

\[
C(x) = \text{Fixed Cost} + \text{Marginal Cost} \times x
\]
Substitute the given values:
\[
C(x) = 1300 + 1600x
\]

### 2. **Using the Cost Function to Answer the Questions**

#### (a) **On a given day, what is the cost of manufacturing 3 pianos?**
Substitute $ x = 3 $ into the cost function:
\[
C(3) = 1300 + 1600 \times 3 = 1300 + 4800 = 6100
\]
So, the cost of manufacturing 3 pianos is **$6,100**.

#### (b) **What is the cost of manufacturing the third piano that day?**
The cost of manufacturing the third piano is the difference in cost between producing 3 pianos and producing 2 pianos:
\[
C(3) - C(2) = (1300 + 1600 \times 3) - (1300 + 1600 \times 2)
\]
\[
C(3) - C(2) = (6100 - 4500) = 1600
\]
So, the cost of manufacturing the third piano is **$1,600**.

#### (c) **What is the cost of manufacturing the 11th piano that day?**
The cost of manufacturing the 11th piano is the difference in cost between producing 11 pianos and producing 10 pianos:
\[
C(11) - C(10) = (1300 + 1600 \times 11) - (1300 + 1600 \times 10)
\]
\[
C(11) - C(10) = (19300 - 17700) = 1600
\]
So, the cost of manufacturing the 11th piano is **$1,600**.

#### (d) **What is the variable cost, fixed cost, and marginal cost?**
- **Variable cost**: The part of the total cost that depends on the number of pianos produced. This is $ 1600x $, where $ x $ is the number of pianos.
- **Fixed cost**: The constant cost, which is **$1,300**.
- **

A piano manufacturer has a daily fixed cost of $1,300 and a marginal cost of $1,600 per piano. Find the cost C(x) of manufacturing x pianos in one day. (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? (e) Graph C(x) as a function of x.

Explore how to calculate the cost of manufacturing pianos with a fixed cost of $1,300 and a marginal cost of $1,600 per piano. Learn to use the cost function C(x) = 1300 + 1600x to solve questions related to manufacturing costs, including finding the cost of the 3rd and 11th pianos. Suitable for Grades 9-12.

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