Let's break down the firm's production function and address each question step by step. The production function is given as:

$Y = F(K, L) = K^{0.5} L^{0.5}$

In the short run, capital $K$ is fixed at $K = 4$, and the firm sells output in a perfectly competitive market at a price $P = 10$. The cost of labor $L$ is represented by $w$ (the wage rate), and the cost of capital is represented by $r$ (the rental rate of capital).

(c) Short-run demand for labor

In the short run, $K$ is fixed at 4. Therefore, the production function simplifies to:

$Y = F(4, L) = 4^{0.5} L^{0.5} = 2L^{0.5}$

The firm's revenue is given by:

$\text{Revenue} = P \cdot Y = 10 \cdot 2L^{0.5} = 20L^{0.5}$

The firm's cost in the short run consists only of labor costs, since $K$ is fixed:

$\text{Cost} = w \cdot L$

To find the firm's short-run demand for labor, we need to maximize profit. Profit is defined as:

$\text{Profit} = \text{Revenue} - \text{Cost} = 20L^{0.5} - wL$

To maximize profit, we take the derivative of the profit function with respect to $L$, and set it equal to 0:

$\frac{d}{dL}\left( 20L^{0.5} - wL \right) = 0$

First, compute the derivative:

$\frac{d}{dL}\left( 20L^{0.5} \right) = 10L^{-0.5}$ $\frac{d}{dL}\left( wL \right) = w$

Setting the derivative equal to 0 gives:

$10L^{-0.5} = w$

Solve for $L$:

$L^{-0.5} = \frac{w}{10}$ $L^{0.5} = \frac{10}{w}$ $L = \left( \frac{10}{w} \right)^2$ $L = \frac{100}{w^2}$

Thus, the firm's short-run demand for labor is:

$L = \frac{100}{w^2}$

(d) Short-run profit-maximizing output

From part (c), we know the labor demand $L = \frac{100}{w^2}$. Now we substitute this into the production function to find the short-run output $Y$:

$Y = 2L^{0.5}$

Substitute $L = \frac{100}{w^2}$:

$Y = 2 \left( \frac{100}{w^2} \right)^{0.5}$ $Y = 2 \times \frac{10}{w}$ $Y = \frac{20}{w}$

Therefore, the firm's short-run profit-maximizing output is:

$Y = \frac{20}{w}$

(e) Long-run profit-maximizing output

In the long run, both capital $K$ and labor $L$ are variable. The firm will choose the optimal combination of $K$ and ( L