Please answer my question regarding maths for finance and economics

Question

A firm has using two inputs to produce a single output, Q . The Firm’s production function is Q=X1 1/2 X2 1/2 If input price of X1 and X2 are $2 and $8 per unit, respectively, get the necessary conditions and use it to determine:

(a) Quantity of X1 and X2 that minimizes the cost.

(b) Minimum cost if the firm produces output of 20 units.

Answer 1

Let me attempt an answer. Maybe, it is late and you already know the details, but possibly it is still worth it...

First writing the equations provided by the statements in the problem.

Q = x1^1/2 * x2^1/2 (1) production function

C = 2*x1 + 8*x2 (2) cost function

a) To find the x1 and x2 that minimizes cost, we need to find the point where the isoquant (a production function curve) is tangent to the cost function (budget constraint) line.

(2) can be rewritten as x2 = C/8 - 1/4*x1 => ∂x2 / ∂x1 = -1/4 (3) - slope of the budget line is -1/4.

From (1) x2^1/2 = Q / x1^1/2 => x2 = Q² / x1 => ∂x2 / ∂x1 = -Q² / x1² (4) - MRTS

Also from (1) x1 * x2 = Q² (5)

Equating (3) and (4) => Q² / x1² = 1/4, or x1² = 4*Q² => x1 = 2*Q. From (5) x2 = 1/2 * Q. These should be the x1 and x2 quantities that minimize the cost.

b) If Q = 20 units, then the cost will be from (2) C = 2*x1 + 8*x2 and replacing x1 and x2 with the values found at point a), 2 * 2 * 20 + 8 * 1/2 * 20 (Q=20). So the cost will be 160.