A shop has a total cost function that is given by the equation TC = 218 + 3Q,

while the demand function is given by the equation P = 107 - 2Q.

Calculate the LOWER quantity of the break even point.

(Notes: Use 3 digits after decimal point)

To find the break-even point, we need to determine when the total revenue equals the total cost. Here's how we proceed step-by-step:

1. **Identify the Total Cost (TC) and Total Revenue (TR) Functions:**

   - Total Cost (TC) function:  
     \[
     TC = 218 + 3Q
     \]
   - Demand (price) function:  
     \[
     P = 107 - 2Q
     \]

2. **Calculate the Total Revenue (TR):**

   Total Revenue (TR) is calculated as Price (P) multiplied by Quantity (Q):  
   \[
   TR = P \times Q = (107 - 2Q) \times Q
   \]

   Simplify this:  
   \[
   TR = 107Q - 2Q^2
   \]

3. **Find the Break-Even Point by Setting TR Equal to TC:**

   Set the Total Revenue equal to the Total Cost:  
   \[
   107Q - 2Q^2 = 218 + 3Q
   \]

4. **Rearrange the Equation into Standard Quadratic Form:**

   Move all terms to one side of the equation:  
   \[
   -2Q^2 + 107Q - 3Q - 218 = 0
   \]

   Simplify:  
   \[
   -2Q^2 + 104Q - 218 = 0
   \]

5. **Solve the Quadratic Equation Using the Quadratic Formula:**

   The quadratic formula is given by:  
   \[
   Q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]

   Here, \(a = -2\), \(b = 104\), and \(c = -218\). Plug these values into the formula:

\[
Q = \frac{-104 \pm \sqrt{(104)^2 - 4(-2)(-218)}}{2(-2)}
\]

Let's compute this to find the two break-even points:The two quantities at the break-even points are:

- \( Q_1 = 2.188 \)
- \( Q_2 = 49.812 \)

The lower quantity of the break-even point is:

\[
\boxed{2.188}
\]

Would you like more details on this calculation, or do you have any other questions?

### Related Questions:
1. How do you interpret the economic meaning of the break-even points?
2. What is the revenue at the lower break-even quantity?
3. What would happen to the break-even points if the fixed cost changes?
4. How do changes in price affect the break-even points?
5. Can you calculate the profit at quantities above the higher break-even point?

**Tip:** Always double-check your calculations when solving quadratic equations, especially when dealing with real-world applications!

A shop has a total cost function that is given by the equation TC = 218 + 3Q, while the demand function is given by the equation P = 107 - 2Q. Calculate the LOWER quantity of the break even point. (Notes: Use 3 digits after decimal point)

Calculate the lower quantity of the break-even point for a shop with a total cost function TC = 218 + 3Q and a demand function P = 107 - 2Q. This problem involves quadratic equations and break-even analysis, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation