If `C(x)=5x+450andR(x)=50z-x^(2)`, then break even points are

Options

A. 45 and 10
B. 35 and 10
C. 15 and 10
D. none of the above

Answer

To find the break-even points for the given cost function \( C(x) = 5x + 450 \) and revenue function \( R(x) = 50z - x^2 \), we need to set the cost equal to the revenue and solve for \( x \).

### Step-by-Step Solution:

1. **Set the Cost Function Equal to the Revenue Function**:
\[ C(x) = R(x) \]
This gives us the equation:
\[ 5x + 450 = 50z - x^2 \]

2. **Rearrange the Equation**:
Move all terms to one side of the equation to set it to zero:
\[ x^2 + 5x + 450 - 50z = 0 \]
This can be rewritten as:
\[ x^2 + 5x + (450 - 50z) = 0 \]

3. **Identify Coefficients**:
From the quadratic equation \( ax^2 + bx + c = 0 \), we have:
- \( a = 1 \)
- \( b = 5 \)
- \( c = 450 - 50z \)

4. **Use the Quadratic Formula**:
The solutions for \( x \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (450 - 50z)}}{2 \cdot 1} \]
Simplifying further:
\[ x = \frac{-5 \pm \sqrt{25 - 1800 + 200z}}{2} \]
\[ x = \frac{-5 \pm \sqrt{200z - 1775}}{2} \]

5. **Determine Break-Even Points**:
The break-even points occur at the values of \( x \) obtained from the above formula. The expression under the square root, \( 200z - 1775 \), must be non-negative for real solutions, which means:
\[ 200z - 1775 \geq 0 \implies z \geq \frac{1775}{200} = 8.875 \]

### Final Answer:
The break-even points are given by:
\[ x = \frac{-5 \pm \sqrt{200z - 1775}}{2} \]
for \( z \geq 8.875 \).