Compute total wages and partial-hour payments

You are given a list of workers. Each worker has:

• id (string)
• hourly_rate (non-negative number)
• one work shift for a given day: start_time , end_time (times such as "12:00" , "22:00" in 24-hour format)

Assume times are within the same day and end_time > start_time (no overnight shifts). Pay is proportional to time worked.

Part 1 — Total payroll for full shifts

Write a function that computes the total amount of money paid to all workers for their full shifts.

Input: list of workers

Output: total pay (number)

Part 2 — Payroll up to a cutoff time (partial shifts)

Write a second function to compute how much to pay workers up to a given cutoff time cutoff_time.

Rules:

• If a worker’s shift ends at or before cutoff_time , pay their full shift.
• If a worker’s shift starts at or after cutoff_time , pay 0 .
• If a worker’s shift overlaps the cutoff (e.g., works 12:00–22:00 and cutoff is 17:00 ), include only the portion worked from start_time up to cutoff_time .

Input: list of workers, cutoff_time

Output: total pay up to cutoff (number)

Follow-up — Caching / avoiding recomputation

If this function is called repeatedly for many different cutoff times, describe (at a high level, in code structure or data structures) how you would cache or preprocess information to avoid recomputing from scratch each time.

(You do not need to use any specific library; just clearly define the approach and expected time/space tradeoffs.)