To determine the minimum unit cost for manufacturing airplane engines, we need to minimize the cost function [tex]\( C(x) \)[/tex], defined as follows:
[tex]\[ C(x) = 1.1x^2 - 638x + 106,380 \][/tex]
Here's a detailed step-by-step solution to find the minimum unit cost:
1. Identify the function to minimize:
We are given:
[tex]\[ C(x) = 1.1x^2 - 638x + 106,380 \][/tex]
2. Find the critical points:
To find the minimum, we first take the derivative of [tex]\( C(x) \)[/tex] with respect to [tex]\( x \)[/tex]. This gives us:
[tex]\[ \frac{dC}{dx} = \frac{d}{dx}(1.1x^2 - 638x + 106,380) \][/tex]
[tex]\[ \frac{dC}{dx} = 2.2x - 638 \][/tex]
3. Set the derivative equal to zero to find the critical points:
[tex]\[ 2.2x - 638 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 2.2x = 638 \][/tex]
[tex]\[ x = \frac{638}{2.2} \][/tex]
[tex]\[ x = 290 \][/tex]
4. Evaluate the cost function at the critical point:
We substitute [tex]\( x = 290 \)[/tex] back into the original cost function [tex]\( C(x) \)[/tex]:
[tex]\[ C(290) = 1.1(290)^2 - 638(290) + 106,380 \][/tex]
5. Calculate the minimum unit cost:
[tex]\[ C(290) = 1.1 \times 84100 - 638 \times 290 + 106,380 \][/tex]
[tex]\[ C(290) = 92,510 - 185,020 + 106,380 \][/tex]
[tex]\[ C(290) = 13870 \][/tex]
Therefore, the minimum unit cost is:
[tex]\[
\boxed{13870}
\][/tex]
