A piano manufacturer has a daily fixed cost of $1,600 and a marginal cost of $1,900 per piano. Find the cost

*C*(*x*)

of manufacturing *x* pianos in one day.

*C*(*x*) =

1,600+1,900x

**Nicely done.**

Use your function to answer the following questions.

(a)

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

$   **Fantastic job!**

(b)

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

$   **Nicely done!**

(c)

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

Given the cost function for manufacturing $ x $ pianos in one day:

\[
C(x) = 1600 + 1900x
\]

We can use this function to answer the following questions:

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

To find the cost of manufacturing 3 pianos, substitute $ x = 3 $ into the cost function:

\[
C(3) = 1600 + 1900(3) = 1600 + 5700 = 7300
\]

**The cost of manufacturing three pianos is** $ \mathbf{7300} $ **dollars**.

### (b) What is the cost of manufacturing the third piano that day?

The cost of manufacturing the third piano is the difference between the cost of making 3 pianos and the cost of making 2 pianos:

\[
C(3) - C(2)
\]

First, find $ C(2) $:

\[
C(2) = 1600 + 1900(2) = 1600 + 3800 = 5400
\]

Now, find the difference:

\[
C(3) - C(2) = 7300 - 5400 = 1900
\]

**The cost of manufacturing the third piano is** $ \mathbf{1900} $ **dollars**.

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

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

\[
C(11) - C(10)
\]

First, find $ C(11) $ and $ C(10) $:

\[
C(11) = 1600 + 1900(11) = 1600 + 20900 = 22500
\]

\[
C(10) = 1600 + 1900(10) = 1600 + 19000 = 20600
\]

Now, find the difference:

\[
C(11) - C(10) = 22500 - 20600 = 1900
\]

**The cost of manufacturing the 11th piano is** $ \mathbf{1900} $ **dollars**.

Would you like further details on any of these calculations or have any additional questions?

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Here are 5 related questions you might find interesting:

1. How does the cost function change if the fixed cost increases?
2. What is the total cost of manufacturing 20 pianos in a day?
3. How would the cost function change if the marginal cost per piano decreases?
4. What is the break-even point if each piano sells for $2500?
5. How would you represent the profit function based on this cost function?

**Tip:** Understanding how to calculate the marginal cost can help in optimizing production strategies and pricing.

Explore the cost of manufacturing pianos with a detailed explanation of fixed costs and marginal costs. Answer key questions about daily manufacturing costs based on a given cost function. Suitable for Grades 9-12.

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