Question text 15Marks The number of shopping trolleys available in the trolley bay at a supermarket has been found to be modelled by an ODE: [math: dTdt=0.006(120−TN2)(cos(πt8+0.1))(t4−47t3+642t2−2160t)]\frac{dT}{dt}=0.006\left(\frac{\sqrt{120-T}}{N^{2}}\right)\left(cos\left(\frac{\pi t}{8}+0.1\right)\right)\left(t^{4}-47t^{3}+642t^{2}-2160t\right) Where: T is the number of trolleys available t is the time of day, in hours (0 - 24) as a decimal N is the number staff allocated to return trolleys to the bay There are 110 trolleys available at the start of a particular day, and 5 staff rostered on to return them. (a) At what time of the day is the lowest percentage availability to be expected? Give your answer to the nearest minute. Give your answer to as 24-hr time e.g. "20:13" would be 8:13pm.Time = Answer 4[input] (b) How many staff are needed (minimum) to ensure that there is always at least 1 trolley available?Staff needed = Answer 5[input]----------------------------Checks: (Part a) At 9am, with 5 staff, there will be approximately 98 trolleys available (Predicted value = 97.915)(Part b) With only 2 staff, there will be no trolleys available by 8:26amNotes Report question issue Question 3 Notes

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Restating the problem in my own words helps clarify what's being asked:
- We have a model for the number of shopping trolleys T as a function of time t (in hours, 0 to 24) with parameters including N (staff allocated to return trolleys). The differential equation is dT/dt = 0.006 * (sqrt(120 − T) / N^2) * cos(πt/8 + 0.1) * (t^4 − 47t^3 + 642t^2 − 2160t). There are 110 trolleys available at the start and 5 staff rostered on to return them.
- Part (a) asks for the time of day when the percentage availability (i.e., T as a fraction of 110) is at its lowest, to the nearest minute, in 24-hour time.
- Part (b) asks for the minimum number of staff N needed to ensure at least 1 trolley is always available (i.e., T < 110 for all times, or equivalently 110 − T > 0 for all t).
- The provided checks indicate that with 5 staff at 9am, approximately 97.915 trolleys are predicted to be available; with 2 staff there would be no trolleys available by 8:26am.

Now, I’ll analyze the given options conceptually for each part.

Part (a): Time of day when lowest percentage availability occurs
- Option: 11:36 (time) – If this value is the expected lowest annualized availability time, we would interpret it as the point in the 24-hour day where dT/dt is most negative when T is not yet at the minimum bound and the cosine term cos(πt/8 + 0.1) and the polynomial factor (t^4 − 47t^3 + 642t^2 − 2160t) cooperate to drive T downward the most. The cosine term introduces an oscillatory component with a period of 16 hours (since argument is πt/8), so around mid-morning to early afternoon, depending on phase, the rate of change could be strongly negative, potentially pushing T lower relative to 110. The check mentioning 9am with 5 staff yields about 97.9 trolleys (which is around 110 − 12.1), suggesting significant depletion occurs in the morning hours but not necessarily the absolute minimum time, unless the dynamics later drive T even lower. Thus 11:36 could plausibly be near a trough if the combined factors align to reduce T further after 9am, but we would need the actual numerical trajectory to confirm.
- Conceptual caveat: The differential equation includes a non-linear term involving T (sqrt(120 − T)) and a time-varying cos factor, so the minimum percentage availability does not occur at a constant rate and depends on the instantaneous balance of depletion (driven by the polynomial in t) and replenishment (through the cos term and the N parameter). Therefore, without performing the numeric integration or plotting, we cannot conclusively select 11:36 as the exact lowest point, though it could be reasonable if the trough of the cosine aligns with a deep point in the t-polynomial near that time.

Part (b): Minimum staff to ensure at least 1 trolley is always available
- Option: 3 (staff) – The checks indicate that with only 2 staff there will be no trolleys available by 8:26am. This strongly suggests that N must be at least 3 or more to maintain T above 0 (i.e., ensure some trolley inventory is always present at all times). Increasing N reduces the depletion rate by dividing by N^2 in the differential equation, effectively slowing the decline of T. If with N = 3 the model still allows T to dip but not reach zero for all times, then 3 could be the minimum threshold. However, if even N = 3 allows a momentary depletion to zero later in the day due to the time-varying cos factor and the t-dependent polynomial, then a larger N would be required. The given check does not specify the exact time when T hits zero for N = 3, only that with 2 staff it hits zero by 8:26am. Therefore, 3 is plausible as the minimum, but strictly speaking we would need to verify the trajectory to ensure T never reaches 0 for all t in 0–24 hours at N = 3.

Synthesis of the analysis across options:
- For Part (a), the time 11:36 could correspond to a trough in the T(t) trajectory depending on the phase of the cosine term and the t-polynomial, but without numeric computation or a plot, we cannot guarantee it is the exact lowest percentage point. The nonlinearity and time-varying factors imply that the absolute minimum is data-driven rather than deducible by inspection alone.
- For Part (b), the observation that 2 staff leads to zero trolleys by 8:26am strongly indicates that the minimum required staff is more than 2. The candidate value of 3 is consistent with the idea of a minimal increase to prevent total depletion, yet this too would hinge on the detailed trajectory; if T ever hits zero with N = 3, then a larger N would be necessary. The note in the checks that 2 staff causes depletion early strongly supports the need for at least 3 staff, with 3 being a reasonable minimum candidate.

In summary, the provided values (11:36 for part a and 3 for part b) align with the qualitative behavior described in the checks and the structure of the differential equation, but precise validation would require solving or simulating the ODE numerically to verify the exact times and minimum N, respectively. The reasoning above outlines the factors and potential pitfalls that determine why those particular numbers might be chosen, and why small changes in N or phase alignment could shift the results substantially.

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