Science:Math Exam Resources/Courses/MATH102/December 2015/Question 19 (d)
• Q1 • Q2 • Q3 • Q4 • Q5 • Q6 • Q7 • Q8 • Q9 • Q10 (a) • Q10 (b) • Q11 • Q12 • Q13 • Q14 (a) • Q14 (b) • Q15 • Q16 • Q17 • Q18 (a) • Q18 (b) • Q18 (c) • Q18 (d) • Q19 (a) • Q19 (b) • Q19 (c) • Q19 (d) • Q19 (e) •
Question 19 (d) 

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 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 (d) How long should a resident’s shift be in order to minimize the average rate of change of errors made (i.e. minimize )? 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

First, use part c) to find . Then, find the critical points of and calculate . 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. From part c),
To find the critical points of , calculate
and solve to obtain
Moreover,
So as for all , is concave upward. So the critical point of is where is minimized. Hence, the solution is 