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.)

### Step 1: Derive the Cost Function $C(x)$

The cost function is generally given by:

\[
C(x) = \text{fixed cost} + (\text{marginal cost}) \times x
\]

Where:
- The **fixed cost** is the cost that does not change regardless of how many pianos are produced. In this case, the fixed cost is \$1,300 per day.
- The **marginal cost** is the cost of producing each additional piano, which is \$1,600 per piano.

Thus, the cost function $C(x)$ for manufacturing $x$ pianos in one day is:

\[
C(x) = 1300 + 1600x
\]

### Step 2: Answer the Questions

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

We substitute $x = 3$ into the cost function:

\[
C(3) = 1300 + 1600(3) = 1300 + 4800 = 6100
\]

So, the cost of manufacturing three 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 total cost between producing three pianos and producing two pianos. We calculate:

\[
C(3) - C(2) = \left(1300 + 1600(3)\right) - \left(1300 + 1600(2)\right) = 6100 - 4500 = 1600
\]

Thus, the cost of manufacturing the third piano is **\$1,600**.

#### (c) What is the cost of manufacturing the 11th piano that day?

Similarly, the cost of manufacturing the 11th piano is the difference between producing 11 pianos and 10 pianos:

\[
C(11) - C(10) = \left(1300 + 1600(11)\right) - \left(1300 + 1600(10)\right) = 19300 - 17700 = 1600
\]

So, the cost of manufacturing the 11th piano is also **\$1,600**.

#### (d) Variable cost, Fixed cost, and Marginal cost

- **Variable cost** refers to the part of the cost that depends on the number of pianos produced. For each piano, the variable cost is $1600x$, since it's tied directly to the number of pianos produced.
  
- **Fixed cost** is the daily fixed cost of \$1,300.
  
- **Marginal cost** is the cost of producing one more piano, which is \$1,600 per piano.

### (e) Graph $C(x)$

The function $C(x) = 1300 + 1600x$ is a straight line with:
- Y-intercept = 1300 (the fixed cost),
- Slope = 1600 (the marginal cost per piano).

The graph will be a linear function that starts at $(0, 1300)$ and increases by 1600 units for each additional piano produced.

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Would you like further details or have any questions about this? Here are five related questions to explore:

1. How does the marginal cost affect the total cost function graph?
2. What happens to the total cost if the fixed cost changes but the marginal cost stays the same?
3. How would the total cost change if the marginal cost were reduced by \$200 per piano?
4. What would the cost be if no pianos were manufactured in a day?
5. How does the slope of the cost function relate to real-world changes in production cost?

**Tip:** When working with cost functions, always differentiate between fixed costs (which do not vary with production) and variable costs, which depend on output.

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 as a function of x.

Explore the cost function for manufacturing pianos, including fixed costs of $1,300 and a marginal cost of $1,600 per piano. This detailed solution explains the calculation of manufacturing costs for multiple pianos, the variable and fixed costs, and includes a graph of the linear cost function. Suitable for high school students learning linear functions and cost analysis.

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