Let's solve it step by step: --- ## **Step 1: Set up the production function with given values** The production function is: \[ q = 10L^{.5}K^{.5} \] - Initial output \( q = 140 \) - Cost-minimizing quantities: \( L = 14 \), \( K = 14 \) Let's **verify** this with the production function: \[ 140 = 10 \times (14^{.5}) \times (14^{.5}) \\ 140 = 10 \times 14 = 140 \text{ (Correct)} \] --- ## **Step 2: Find required labor for new output (holding capital constant)** Now, \( q = 280 \) and \( K \) remains at 14 (fixed in the short run). Plug into the production function: \[ 280 = 10L^{.5}(14^{.5}) \] \[ 280 = 10L^{.5} \times 3.7417 \] \[ 280 = 37.417 L^{.5} \] \[ L^{.5} = \frac{280}{37.417} = 7.486 \] \[ L = (7.486)^2 = 56.04 \] **Explanation:** We set capital at 14, substitute into the production function, solve for \( L \), and square to get the required amount. --- ## **Final Answer** \[ \boxed{56.04} \] The firm will now require **56.04 units of labor** to produce 280 units of output, holding capital fixed at 14 units.