Final answer:
To minimize total costs, use the method of Lagrange multipliers and calculate the optimal allocation of eligible people to each scheme. We should place approximately 1662 eligible people in scheme A.
Step-by-step explanation:
To minimize the total costs, we can use the method of Lagrange multipliers to allocate the eligible people to each scheme. Let's define a Lagrange function:
L(x, y, λ) = TCA - λ(Total eligible people - x - y) - TCB
Taking the partial derivatives of the Lagrange function with respect to x, y, and λ, we can set them equal to zero:
dL/dx = 10 + 1.2x - λ = 0
dL/dy = 5 + 1y - λ = 0
dL/dλ = Total eligible people - x - y = 0
Solving these equations simultaneously, we can determine the optimal values for x and y. Let's calculate:
10 + 1.2x - λ = 0
5 + 1y - λ = 0
Total eligible people - x - y = 0
Substituting the third equation into the first two equations, we get:
10 + 1.2x - (Total eligible people - x - y) = 0 => 2x + y = Total eligible people - 10
5 + 1y - (Total eligible people - x - y) = 0 => x + 2y = Total eligible people - 5
Solving these equations, we can find the optimal values for x and y:
x = (Total eligible people - 15) / 3
y = (Total eligible people - 5) / 3
Now, we are given that the total eligible people is 5,000. Substituting this value into the equations, we can determine the optimal values for x and y:
x = (5000 - 15) / 3
≈ 1662
y = (5000 - 5) / 3
≈ 1665
Rounded to the nearest integer, we should place approximately 1662 eligible people in scheme A.
Learn more about Allocation of resources here: