To whoever is able to help out, thank you so much!

. 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 ndings (source). In this problem, you will consider a mathematical model of the problem to learn what inuences 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) = i iIf2 + 2. 250 50 (a) What are the units of E'(t)? (b) What is the model domain of E'(t)? That is, which values of t makes sense in our hospital model? (c) When is E'(t) increasing? decreasing? For each region of increase/decrease, suggest a plausible explanation for why the error rate is going up or down. (d) 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). (e) What is the total number of errors, E (if), made t hours into a shift? (f) What is the average rate of change of E (it) from the start of a shift (if = 0) up until time t? Call it AGE). (g) Explain, as you would to a hospital administrator, why it makes sense to minimize AGE) rather than EU). (h) How long should a resident's shift be in order to minimize AH)? Call this value t*. (i) Sketch E(t). Label any minima, maxima and/or inection points. On the same axes, draw a line tangent to EU?) at 15*. Label the coordinates of the intersection (be careful that your scales for the two functions match). (j) Does your expectation from part b match your result in part e? There is something arguably counterintuitive about this optimum. Explain why the actual minimum occurs where it does (with regard to the regions of increase and decrease in E'(t))