
Concept explainers
In this application, we set up a mathematical model for determining the total costs in setting up a training program, such as a hospital might use. Then we use calculus to find the time interval between training programs that produces the minimum total cost. The model assumes that the demand for trainees is constant and that the fixed cost of training a batch of trainees is known. Also, it is assumed that people who are trained, but for whom no job is readily available, will be paid a fixed amount per month while waiting for a job to open up.
The model uses the following variables.
D = demand for trainees per month
N = number of trainees per batch
m = time interval in months between successive batches of trainees
t = length of training program in months
The total cost of training a batch of trainees is given by
After training, personnel are given jobs at the rate of D per month. Thus,
While the costs during the second month are
Which can be factored to give
The expression in bracket is the sum of the terms of an arithmetic sequence, discussed in most algebra texts. Using formulas for arithmetic sequences, the expression in brackets can be shown to equal
As the total cost for keeping jobless trainees.
The total cost per batch is the sum of the training cost per batch.
Source: P.I., Goyal and S.K. Goyal.
Find

To find:
The expression
Answer to Problem 1EA
Solution:
The expression
Explanation of Solution
Given:
The given function is:
Approach:
Differentiate the given function with respect to m.
Calculation:
Consider the given function,
Differentiate with respect to m:
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Chapter 6 Solutions
Calculus For The Life Sciences
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