https://r2.erweima.ai/mathbot/image/1727062016140hxvi6y98.jpg

The problem in the image presents a production cost function $ C(n) = 200 - 10n + 10n^2 $, where $ C(n) $ represents the cost in dollars to produce $ n $ toasters. 

We are given two pieces of information:
1. The cost of producing 10 toasters is $1000.
2. It is false that the cost of producing 20 toasters is $4000.

Let's analyze these claims by substituting $ n = 10 $ and $ n = 20 $ into the function:

### First, for $ n = 10 $:
\[
C(10) = 200 - 10(10) + 10(10)^2
\]
\[
C(10) = 200 - 100 + 10(100)
\]
\[
C(10) = 200 - 100 + 1000 = 1100
\]
This contradicts the statement that the cost of producing 10 toasters is $1000, as the function gives $1100 instead.

### Now, for $ n = 20 $:
\[
C(20) = 200 - 10(20) + 10(20)^2
\]
\[
C(20) = 200 - 200 + 10(400)
\]
\[
C(20) = 200 - 200 + 4000 = 4000
\]
This confirms that the cost of producing 20 toasters is $4000, contradicting the claim that this is false.

### Conclusion:
- The function does **not** correctly give the cost of producing 10 toasters as $1000 (it gives $1100 instead).
- It **does** give the cost of producing 20 toasters as $4000, which conflicts with the statement that this is false.

Would you like further clarification on this or have additional questions? Here are some related questions for deeper understanding:

1. How would you interpret the quadratic nature of the cost function?
2. What would be the cost for producing 15 toasters using the given function?
3. Can the cost function be modified to meet the conditions given for 10 toasters?
4. How does the $ n^2 $ term in the cost function affect larger production scales?
5. Could external factors explain the discrepancy in the cost for 10 toasters?

**Tip:** To verify claims about functions, always substitute the values into the function and calculate the results carefully.

Es verdad que el costo de producir '10' tostadoras en dólares, es 1000 y es falso que el costo de producir '20' tostadoras en dólares es 4000, en la siguiente función de producción: C(n) = 200 - 10n + 10n^2.

Analyze the cost of producing 10 and 20 toasters using the quadratic cost function C(n) = 200 - 10n + 10n^2. Verify the cost statements and explore the algebraic solution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation