Question

A defense contractor is starting production on a new missile control system. On the basis of data collected during the assembly of the first 16 control systems, the production manager obtained the following function describing the rate of labour use: L ′ (x)=2400x −1/4 2/2 For example, after assembly of 16 units, the rate of assembly is 1200 labor hours per unit. (a) What is the rate of assembly after assembly of 81 units? (b) If 30000 labor hours are required to assemble the first 16 units, how many labor hours L(x) will be required to assemble the first x units? (c) What will labor hour be for the first 81 units?

Answer 1

(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.