Find the break even points when, `p= 72-4x, C(x) = 16x + 180`

Answer

To find the break-even points where total revenue equals total cost, we will follow these steps:

### Step 1: Define the Revenue and Cost Functions
The price function is given as:
\[ P = 72 - 4x \]

The cost function is given as:
\[ C(x) = 16x + 180 \]

### Step 2: Write the Total Revenue Function
Total revenue (R) is calculated as the product of price (P) and quantity (x):
\[ R(x) = P \cdot x = (72 - 4x) \cdot x \]
Expanding this:
\[ R(x) = 72x - 4x^2 \]

### Step 3: Set Revenue Equal to Cost
To find the break-even points, we set the total revenue equal to the total cost:
\[ R(x) = C(x) \]
Thus,
\[ 72x - 4x^2 = 16x + 180 \]

### Step 4: Rearrange the Equation
Rearranging the equation gives us:
\[ 72x - 4x^2 - 16x - 180 = 0 \]
Combining like terms:
\[ -4x^2 + 56x - 180 = 0 \]
Multiplying through by -1 to simplify:
\[ 4x^2 - 56x + 180 = 0 \]

### Step 5: Simplify the Quadratic Equation
Dividing the entire equation by 4:
\[ x^2 - 14x + 45 = 0 \]

### Step 6: Factor the Quadratic Equation
We can factor the quadratic:
\[ x^2 - 9x - 5x + 45 = 0 \]
Grouping the terms:
\[ (x - 9)(x - 5) = 0 \]

### Step 7: Solve for x
Setting each factor to zero gives us:
1. \( x - 9 = 0 \) → \( x = 9 \)
2. \( x - 5 = 0 \) → \( x = 5 \)

### Conclusion
The break-even points are:
\[ x = 5 \text{ and } x = 9 \]