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Answer :
Final answer:
To minimize the cost of producing 570 units, the production should be distributed as follows: Factory 1 should produce 285 units, while Factories 2 and 3 should each produce 142.5 units. The minimum cost for producing 570 units is $1,204,000.
Explanation:
To determine how the production should be distributed to minimize the cost of an order of 570 units, we need to find the values of x, y, and z that will minimize the total cost function. The total cost for factory 1 is given by C1(x) = x^2 + 60000, where x represents the number of units produced. The total cost for factories 2 and 3 are given by C2(y) = (4y)^2 + 20000 and C3(z) = (3z)^2 + 30000, where y and z represent the number of units produced by factories 2 and 3 respectively.
To minimize the total cost, we need to minimize the sum of these three cost functions. Let's denote the total cost as C(x, y, z).
C(x, y, z) = C1(x) + C2(y) + C3(z)
Substituting the given cost functions:
C(x, y, z) = x^2 + 60000 + (4y)^2 + 20000 + (3z)^2 + 30000
Now, we need to find the values of x, y, and z that minimize C(x, y, z) subject to the constraint that the total number of units produced is 570.
x + y + z = 570
To solve this problem, we can use the method of Lagrange multipliers. However, since this is a High School level question, we will use a simpler approach.
Let's solve the constraint equation for one variable and substitute it into the cost function:
x = 570 - y - z
C(y, z) = (570 - y - z)^2 + 60000 + (4y)^2 + 20000 + (3z)^2 + 30000
Expanding and simplifying the cost function:
C(y, z) = 2y^2 + 2z^2 + 8yz - 1140y - 1140z + 204000
To minimize C(y, z), we need to find the values of y and z that make the partial derivatives of C(y, z) with respect to y and z equal to zero:
dC/dy = 4y + 8z - 1140 = 0
dC/dz = 4z + 8y - 1140 = 0
Solving these equations simultaneously, we find y = 142.5 and z = 142.5.
Substituting these values back into the constraint equation, we can solve for x:
x = 570 - y - z
= 570 - 142.5 - 142.5
= 285
Therefore, to minimize the cost of producing 570 units, the production should be distributed as follows:
- Factory 1: 285 units
- Factory 2: 142.5 units
- Factory 3: 142.5 units
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