Practice problem

Integrating to Find Net or Accumulated Change

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Contributors

Joshua Siktar
Liza Sulkin

A picture of a crowded restaurant — Image by RonPorter via https://pixabay.com/en/restaurant-cafe-einstein-restaurant-563127/

Hayley is starting work as a waitress at a restaurant in her hometown. She knows that the restaurant opens at $11$ AM (time $t = 0$ hours) and closes at $11$ PM (time $t = 12$ hours) every day.

The number of occupied tables in the restaurant is modeled as a function of time: $T(t) = -\frac{11}{32}t^2 + \frac{33}{8}t$ .

In selecting a shift, Hayley's strategy is simple: pick a block of time with a lot of customers to maximize the amount of money she earns in tips.

If her manager asks her to select a single, continuous $4$ -hour shift between $11$ AM and $11$ PM, what shift should she select to maximize the amount of tables that are occupied while she works?