Final answer:
The optimal quantity of labor and capital to minimize costs while producing 64 units of output is 64 units of labor and 1 unit of capital.
Step-by-step explanation:
The optimal quantity of labor and capital to minimize costs while producing 64 units of output can be found by determining the input combination that minimizes total cost. To do this, we need to calculate the marginal product of labor (MPL) and the marginal product of capital (MPK) at each level of labor. The MPL can be calculated by taking the derivative of the production function with respect to labor, and the MPK can be calculated by taking the derivative of the production function with respect to capital.
Using the production function q = KL, we can calculate the MPL as dQ/dL = K and the MPK as dQ/dK = L. Since Q = 64 and K = $1, we can determine the value of L that satisfies the equation 64 = KL. Solving for L, we get L = 64/K = 64/1 = 64 units of labor. Therefore, the firm should hire 64 units of labor and use 1 unit of capital to produce 64 units of output at the minimum cost.