# OSH/4

From UBCMATH WIKI

< OSH

- The length of shifts worked by medical residents and other hospital staff is always of great public concern. CBC's Marketplace recently investigated the issue and published their findings. In this OSH, you will consider a mathematical model of the problem to learn what influences the optimal choice of shift length for public safety.

In deciding how long a resident's shift in the emergency room should be, the Chief of Staff at Vancouver General Hospital would like to minimize the average rate at which errors are made. Let $E(t)$ be the number of errors made by a resident from the start of a shift until $t$ hours into the shift. The instantaneous rate of change of errors made is $E'(t)=4-t+\frac{1}{16}t^2$.- During what interval of time is $E'(t)$ increasing? decreasing? For each region of increase/decrease, suggest an explanation for why the error rate is going up or down.
- To build an expectation of the formal results to come, state whether you expect the optimal choice of shift length to come before, at or after the minimum of $E'(t)$ (this expectation will not be evaluated for correctness - the point is to encourage you to establish an expectation).
- What is the total number of errors, $E(t)$, made $t$ hours into a shift?
- What is the average rate of change of $E(t)$ from the start of a shift ($t=0$) up until time $t$? Call it $A(t)$.
- How long should a resident's shift be in order to minimize the average rate of change of errors made (i.e. minimize $A(t)$)?
- Sketch $E(t)$. Label any minima, maxima and/or inflection points. On the same axes, draw a line that shows when the average error rate is minimized. Label any important points on this line.
- Does your expectation from part b match your result in part e? There is something arguably counter-intuitive about this optimum. Explain why the actual minimum occurs where it does (with regard to the regions of increase and decrease in $E'(t)$).

- This question will deal with the aftermath of the midterm. It is extremely important for you to identify and address your mistakes on the midterm before they hinder your further performance in the course. Download the solutions from the Midterm information page and carefully check that your exam was graded according to the rubric. Next, for each question you got wrong, carefully go through it and characterize each, and every, mistake you made as:
- A:
**A**lgebra or**A**rithmetic error - B: not knowing how to
**B**egin the problem, not understanding what the question is asking for - C: not understanding a
**C**oncept or how to apply it in this particular context (specify symbolic, graphic, or geometric) - D:
**D**istracted or careless mistake (rushed) - E: other reasons (please specify)

- A: