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!