(a) The function given is used to describe the rate of labor use. So, to find the rate of assembly after 81 units, we need to substitute x=81 in L'(x) = 2400x^(-1/4). After doing the substitution:
L'(81) = 2400*(81^(-1/4)) = 800.0 labor hours per unit
This means, after assembly of 81 units, the rate of assembly will be 800 labor hours per unit.
(b) The function L(x), which describes the total labor hours needed to assemble the first x units can be found by integrating the function L'(x). So, we have to integrate 2400x^(-1/4) w.r.t 'x'.
Integration[2400x^(-1/4)] dx = 3200*x^(3/4)
Thus, the function L(x) = 3200*x^(3/4) shows the total labor hours to assemble the first x units.
(c) Finally, to find the total labor hours required to assemble the first 81 units, we need to substitute x=81 into the function L(x).
L(81) = 3200*(81^(3/4)) = 86,400 labor hours.
So, 86,400 labor hours are needed to assemble the first 81 units.