To determine the minimum unit cost [tex]\( C(x) \)[/tex] for manufacturing airplane engines, we need to analyze the given cost function:
[tex]\[ C(x) = 0.9 x^2 - 306 x + 36,001 \][/tex]
This function is a quadratic equation [tex]\( C(x) = ax^2 + bx + c \)[/tex], where [tex]\( a = 0.9 \)[/tex], [tex]\( b = -306 \)[/tex], and [tex]\( c = 36,001 \)[/tex].
The graph of a quadratic function is a parabola. Since the coefficient [tex]\( a \)[/tex] (0.9) is positive, the parabola opens upward, meaning the vertex of the parabola represents the minimum point.
To find the number of engines [tex]\( x \)[/tex] that results in the minimum cost, we can use the vertex formula for a quadratic function, where the x-coordinate of the vertex is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Substitute the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{-306}{2 \times 0.9} \][/tex]
[tex]\[ x = \frac{306}{1.8} \][/tex]
[tex]\[ x = 170 \][/tex]
Next, substitute [tex]\( x = 170 \)[/tex] back into the cost function [tex]\( C(x) \)[/tex] to find the minimum unit cost:
[tex]\[ C(170) = 0.9 (170)^2 - 306 (170) + 36,001 \][/tex]
Calculate [tex]\( (170)^2 \)[/tex]:
[tex]\[ 170^2 = 28,900 \][/tex]
Now substitute and solve the equation step by step:
[tex]\[ C(170) = 0.9 \times 28,900 - 306 \times 170 + 36,001 \][/tex]
[tex]\[ C(170) = 26,010 - 52,020 + 36,001 \][/tex]
[tex]\[ C(170) = 26,010 + 36,001 - 52,020 \][/tex]
[tex]\[ C(170) = 62,011 - 52,020 \][/tex]
[tex]\[ C(170) = 9,991 \][/tex]
Therefore, the minimum unit cost is:
[tex]\[ \boxed{9,991} \][/tex]
So, the minimum unit cost to manufacture airplane engines is \$9,991.
