A vehicle factory manufactures cars. The unit cost \( C \) (the cost in dollars to make each car) depends on the number of cars made. If \( x \) cars are made, then the unit cost is given by the function \( C(x)=0.5 x^{2}-100 x+24,029 \). What is the minimum unit cost? Do not round your answer.

To find the minimum unit cost given by the function \( C(x) = 0.5x^2 - 100x + 24,029 \), we can use the vertex formula for a quadratic function. The vertex of a parabola described by the equation \( ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \). In our case, we have: - \( a = 0.5 \) - \( b = -100 \) Now, we can calculate the value of \( x \) at which the minimum unit cost occurs: \[ x = -\frac{-100}{2 \cdot 0.5} = \frac{100}{1} = 100 \] Next, we substitute \( x = 100 \) back into the cost function \( C(x) \) to find the minimum unit cost: \[ C(100) = 0.5(100)^2 - 100(100) + 24,029 \] Calculating each term: 1. \( 0.5(100)^2 = 0.5 \cdot 10,000 = 5,000 \) 2. \( -100(100) = -10,000 \) 3. The constant term is \( 24,029 \) Now, we can combine these results: \[ C(100) = 5,000 - 10,000 + 24,029 \] \[ C(100) = -5,000 + 24,029 \] \[ C(100) = 19,029 \] Thus, the minimum unit cost is \[ \boxed{19029} \]