Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To find the minimum unit cost for manufacturing airplane engines, we need to determine the minimum value of the given quadratic cost function [tex]\( C(x) = 0.7x^2 - 322x + 55046 \)[/tex].
Quadratic functions of the form [tex]\( C(x) = ax^2 + bx + c \)[/tex] open upwards (and therefore have a minimum point) when the coefficient of [tex]\( x^2 \)[/tex] (denoted as [tex]\( a \)[/tex]) is positive. Here, [tex]\( a = 0.7 \)[/tex], which is positive, confirming that the parabola opens upwards.
The vertex of a parabola [tex]\( ax^2 + bx + c \)[/tex] occurs at the value of [tex]\( x \)[/tex] given by:
[tex]\[ x = \frac{-b}{2a} \][/tex]
For the function [tex]\( C(x) = 0.7x^2 - 322x + 55046 \)[/tex]:
- [tex]\( a = 0.7 \)[/tex]
- [tex]\( b = -322 \)[/tex]
Substitute these values into the vertex formula to find [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-(-322)}{2 \cdot 0.7} = \frac{322}{1.4} = 230 \][/tex]
So, the number of engines that minimizes the unit cost is [tex]\( x = 230 \)[/tex].
Next, we substitute [tex]\( x = 230 \)[/tex] back into the cost function [tex]\( C(x) \)[/tex] to find the minimum unit cost:
[tex]\[ C(230) = 0.7(230)^2 - 322(230) + 55046 \][/tex]
Let's break this calculation down step-by-step:
1. Calculate [tex]\( (230)^2 \)[/tex]:
[tex]\[ 230^2 = 52900 \][/tex]
2. Multiply [tex]\( 0.7 \)[/tex] by [tex]\( 52900 \)[/tex]:
[tex]\[ 0.7 \times 52900 = 37030 \][/tex]
3. Multiply [tex]\( 322 \)[/tex] by [tex]\( 230 \)[/tex]:
[tex]\[ 322 \times 230 = 74060 \][/tex]
4. Substitute these values into the cost function:
[tex]\[ C(230) = 37030 - 74060 + 55046 \][/tex]
5. Simplify this expression:
[tex]\[ C(230) = 37030 - 74060 + 55046 = 18016 \][/tex]
Thus, the minimum unit cost is [tex]\( C(230) = 18015.999999999993 \)[/tex], which can be approximated to [tex]\( 18016 \)[/tex] without rounding.
Therefore, the minimum unit cost of manufacturing the airplane engines is [tex]\( \$18015.999999999993 \)[/tex] when 230 engines are made.
Quadratic functions of the form [tex]\( C(x) = ax^2 + bx + c \)[/tex] open upwards (and therefore have a minimum point) when the coefficient of [tex]\( x^2 \)[/tex] (denoted as [tex]\( a \)[/tex]) is positive. Here, [tex]\( a = 0.7 \)[/tex], which is positive, confirming that the parabola opens upwards.
The vertex of a parabola [tex]\( ax^2 + bx + c \)[/tex] occurs at the value of [tex]\( x \)[/tex] given by:
[tex]\[ x = \frac{-b}{2a} \][/tex]
For the function [tex]\( C(x) = 0.7x^2 - 322x + 55046 \)[/tex]:
- [tex]\( a = 0.7 \)[/tex]
- [tex]\( b = -322 \)[/tex]
Substitute these values into the vertex formula to find [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-(-322)}{2 \cdot 0.7} = \frac{322}{1.4} = 230 \][/tex]
So, the number of engines that minimizes the unit cost is [tex]\( x = 230 \)[/tex].
Next, we substitute [tex]\( x = 230 \)[/tex] back into the cost function [tex]\( C(x) \)[/tex] to find the minimum unit cost:
[tex]\[ C(230) = 0.7(230)^2 - 322(230) + 55046 \][/tex]
Let's break this calculation down step-by-step:
1. Calculate [tex]\( (230)^2 \)[/tex]:
[tex]\[ 230^2 = 52900 \][/tex]
2. Multiply [tex]\( 0.7 \)[/tex] by [tex]\( 52900 \)[/tex]:
[tex]\[ 0.7 \times 52900 = 37030 \][/tex]
3. Multiply [tex]\( 322 \)[/tex] by [tex]\( 230 \)[/tex]:
[tex]\[ 322 \times 230 = 74060 \][/tex]
4. Substitute these values into the cost function:
[tex]\[ C(230) = 37030 - 74060 + 55046 \][/tex]
5. Simplify this expression:
[tex]\[ C(230) = 37030 - 74060 + 55046 = 18016 \][/tex]
Thus, the minimum unit cost is [tex]\( C(230) = 18015.999999999993 \)[/tex], which can be approximated to [tex]\( 18016 \)[/tex] without rounding.
Therefore, the minimum unit cost of manufacturing the airplane engines is [tex]\( \$18015.999999999993 \)[/tex] when 230 engines are made.
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.