To derive the equilibrium output and equilibrium profit for the monopolist, we need to find the quantity of output (q) that maximizes her profit.

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The monopolist’s profit is given by the difference between total revenue and total cost.

- Total Revenue (TR):

TR = P * q

TR = (85 – 3q) * (2 * x^(1/2))

TR = (85 – 3q) * (2 * x^(1/2))

- Total Cost (TC):

TC = p * x

TC = 5 * x

- Profit (π):

Π = TR – TC

Π = (85 – 3q) * (2 * x^(1/2)) – 5 * x

Now, we want to maximize profit, so we’ll take the derivative of the profit function with respect to q and set it equal to zero to find the equilibrium output:

Dπ/dq = 0

d/dq [(85 – 3q) * (2 * x^(1/2)) – 5 * x] = 0

First, let’s find the derivative of the profit function with respect to q:

Dπ/dq = [2 * x^(1/2)] * (-3) + (85 – 3q) * [2 * (1/2) * x^(-1/2)] = -6x^(1/2) + (85 – 3q) * x^(-1/2)

Now, set it equal to zero and solve for q:

-6x^(1/2) + (85 – 3q) * x^(-1/2) = 0

Multiply through by x^(1/2) to simplify:

-6 + (85 – 3q) = 0

Now, isolate -3q:

-3q = 6 – 85

-3q = -79

Divide by -3:

Q = 79 / 3

Q ≈ 26.33 (rounded to two decimal places)

So, the equilibrium output (q) is approximately 26.33.

To find the equilibrium profit, substitute this value of q back into the profit function:

Π = (85 – 3q) * (2 * x^(1/2)) – 5 * x

Π = (85 – 3 * 26.33) * (2 * x^(1/2)) – 5 * x

Calculate π:

Π ≈ (85 – 79) * (2 * x^(1/2)) – 5 * x

Π ≈ 6 * (2 * x^(1/2)) – 5 * x

Π ≈ 12x^(1/2) – 5x

Now, the monopolist’s equilibrium profit depends on the value of x (the input), which we don’t have information about. Therefore, you would need to know the value of x to calculate the exact equilibrium profit.