Ina owns a firm that transforms labor (N) and capital (K) inputs into product (Y ) according to the Cobb-Douglas production function: Y = zKaN 1a

Ina is short-sighted and makes no new investments, relying on existing capital K = 10 and total factor productivity z = 20 to maximize profits. The share of capital is 0.3 (a = 0.3). Ina can hire any number of workers without affecting the wage (w = $10 per hour) and all the workers are identical to Ina in terms of preferences and constraints and maximize their utility by freely choosing their leisure time.

(i) If N corresponds to working hours per day, how many workers should Ina hire if her workers pay a lump sum tax? How many workers if the workers pay a labor income tax?

(ii) Can Ina's firm really maximize profits?

please show all the work and the labels for the graph