Changing rates at time cutoffs create a piecewise model: each rate applies to its own stretch of minutes. Build each plan’s total from the charges already earned plus the charge for later minutes. Then set the totals equal and use a Desmos regression to find the time. Keep earlier charges instead of restarting the bill at each cutoff. Convert hourly rates to per-minute rates before comparing.
Hints
- Hint 1
A flat fee is paid once. By 120 minutes, Plan A has charged the fee plus one hour at the next rate. For a later time , how many minutes get the final hourly rate?
- Hint 2
Plan B charges its first 80 minutes at the original rate even when the session lasts longer. Only minutes get the later rate. How would you add those two charges?
- Hint 3
Both plans cover the same number of minutes, . Set their totals equal. A Desmos regression can find the value that makes the two expressions match.
Step-by-step
Build the two costs and solve in Desmos
Step 1Build Plan A’s cost after 120 minutes
Let be the total minutes. For , Plan A has charged $50 for the first 60 minutes and $21 for the next 60 minutes, which is one hour. An hour has 60 minutes, so the final hourly rate applies to only the extra minutes:
- Step 2
Build Plan B’s cost after 80 minutes
For , Plan B still charges $0.65 per minute for all first 80 minutes. Only the remaining minutes cost $0.30 each, so its total is
- Step 3
Set the total costs equal
At the same tutoring time , the plans match when . Replace each total with the expression you built:
Don’t apply the later rates to every minute. The earlier charges stay in the total.
- Step 4
Solve and check the time cutoffs
Type in Desmos. Change to and to so Desmos fits the value that makes the charges equal. Under PARAMETERS, it shows . That’s past both cutoffs, 120 and 80 minutes, so the formulas apply. The plans cost the same at 260 minutes. Choice B.