https://file.aiquickdraw.com/mathbot/file/17289341632505jp6uyga.jpg

The image provides a business problem involving revenue, cost, and profit functions. Here's a breakdown of the relevant information:

1. **Revenue Function**:  
   \( R(x) = 70x - 0.003x^2 \), where \( x \) is the number of units.

2. **Cost Function**:  
   \( C(x) = 41 + 60x \).

3. **Profit Function**:  
   \( P(x) = R(x) - C(x) \). Substituting the given functions:
   \[
   P(x) = (70x - 0.003x^2) - (41 + 60x) = -0.003x^2 + 10x - 41
   \]

   This matches the profit function given in the image, so this part is correct.

4. **Break-even Points** (where \( P(x) = 0 \)):  
   To find the break-even points, solve:
   \[
   -0.003x^2 + 10x - 41 = 0
   \]
   Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where:
   - \( a = -0.003 \)
   - \( b = 10 \)
   - \( c = -41 \)

   Let's calculate this step-by-step:

\[
x = \frac{-10 \pm \sqrt{10^2 - 4(-0.003)(-41)}}{2(-0.003)}
\]
\[
x = \frac{-10 \pm \sqrt{100 - 0.492}}{-0.006}
\]
\[
x = \frac{-10 \pm \sqrt{99.508}}{-0.006}
\]
\[
x = \frac{-10 \pm 9.975}{-0.006}
\]
This results in two values:
\[
x_1 = \frac{-10 + 9.975}{-0.006} \approx 4.167
\]
\[
x_2 = \frac{-10 - 9.975}{-0.006} \approx 3329.17
\]

The quadratic formula yields the break-even points as \( x_1 \approx 4.167 \) and \( x_2 \approx 3329.17 \).

### Issue with the submission:
Although the values you entered (4.167, 3329.17) are mathematically correct, they are being flagged as incorrect by the system. The discrepancy could be due to formatting, rounding, or how the answer is expected to be entered.

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How does changing the quadratic coefficients affect the break-even points?
2. Can you explain why a business might have two break-even points in real-world scenarios?
3. What does the large value of \( x_2 = 3329.17 \) signify in terms of profit and production?
4. How would the profit function change if the cost of production per unit increased?
5. What are some common mistakes when solving quadratic equations in business contexts?

**Tip**: Always double-check the format or rounding expectations for submitting answers, as that could be the cause of system errors.

The marketing department estimates the revenue function as R(x) = 70x - 0.003x^2 and the cost function as C(x) = 41 + 60x. Calculate the profit function and solve for the break-even points using the quadratic formula.

Learn how to calculate the profit function for a business and solve for break-even points using the quadratic formula. The problem involves a revenue function R(x) = 70x - 0.003x^2 and a cost function C(x) = 41 + 60x. Ideal for high school algebra students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation