Question: A supply company manufactures copy machines. The unit cost \( C \) (the cost in dollars to make…

A supply company manufactures copy machines. The unit cost \( C \) (the cost in dollars to make each copy machine) depends on the number of machines made. If \( x \) machines are made, then the unit cost is given by the function \( C(x) = 0.7x^2 - 126x + 22,293 \). What is the minimum unit cost?

Do not round your answer.

Unit cost:

Solution

To find the minimum unit cost, we need to determine the vertex of the quadratic function given. The cost function is: \[ C(x) = 0.7x^2 - 126x + 22,293 \] This is a quadratic function in the form \( ax^2 + bx + c \). For a quadratic \( ax^2 + bx + c \), the vertex \( x \)-coordinate is given by: \[ x = -\frac{b}{2a} \] Substitute the coefficients \( a = 0.7 \) and \( b = -126 \) into the formula: \[ x = -\frac{-126}{2 \times 0.7} \] \[ x = \frac{126}{1.4} \] \[ x = 90 \] Now substitute \( x = 90 \) back into the cost function to find the minimum cost: \[ C(90) = 0.7(90)^2 - 126(90) + 22,293 \] \[ C(90) = 0.7(8100) - 11,340 + 22,293 \] \[ C(90) = 5,670 - 11,340 + 22,293 \] \[ C(90) = 16,623 \] The minimum unit cost is $16,623.