Internet Café Charges and the Daily Cap
An internet café charges $4 for the first hour or any part of an hour, then $2.50 for each additional half-hour or part of a half-hour, up to a daily maximum of $19. A student plans to study at the café for 4.5 hours and wants to know the cost before arriving. A friend claims that 'staying 5.5 hours costs only $1.50 more than staying 4.5 hours because the daily cap kicks in.' The student also has exactly $15 and needs to know the longest whole-hour stay that fits the budget, keeping in mind that any part of a half-hour is charged as a full half-hour. The café's posted rules do not mention prorating. The student is deciding between this café and a library room that is free but closes at 5 p.m.
1. Compute the cost for 4.5 hours and for 5.5 hours, then evaluate the friend's claim.
Read the worked answer
For 4.5 hours: the first hour costs $4. The remaining 3.5 hours is 7 half-hours, because 3.5 divided by 0.5 is 7. Each half-hour costs $2.50, so the additional charge is 7(2.50) = $17.50. The uncapped total is 4 + 17.50 = $21.50. The daily maximum is $19, so the student pays $19.
For 5.5 hours: the remaining 4.5 hours is 9 half-hours. The additional charge is 9(2.50) = $22.50, and the uncapped total is 4 + 22.50 = $26.50. The cap again limits the charge to $19.
Both visits cost $19, so the difference is $0, not $1.50. The friend's claim is false. The cap is the reason the two costs are equal: once the uncapped cost exceeds $19, extra time in the same day does not add to the bill. Without the cap, the difference would have been $5, not $1.50.
2. Using exactly $15 and the café's rounding rule, determine the longest whole-hour stay the student can afford. Show why the next half-hour is not affordable.
Read the worked answer
The first hour costs $4, leaving 15 - 4 = $11 for additional half-hours.
Each half-hour costs $2.50. Four half-hours cost 4(2.50) = $10, so the total is 4 + 10 = $14, which is within $15. Five half-hours cost 5(2.50) = $12.50, so the total is 4 + 12.50 = $16.50, which is over budget.
Four additional half-hours equal 2 hours, so the longest stay is 1 + 2 = 3 hours, costing $14. Any stay over 3 hours, even by one minute, falls into the next half-hour and is charged as a fifth additional half-hour, making the cost $16.50. For example, 3 hours 10 minutes is charged as 1 hour plus 5 half-hours, not 3 hours plus a small fraction. Because the café does not prorate, the student cannot stretch the $15 to 3.5 hours.
3. At what length of stay does the $19 cap first apply, and how does a 6-hour stay compare with a competitor that charges a flat $3.25 per hour with no cap?
Read the worked answer
Before the cap, the cost for h hours, where h is at least 1, is 4 + 2.50(2)(h - 1) because each hour after the first contains two half-hours. That simplifies to 4 + 5(h - 1) = 5h - 1.
Set 5h - 1 = 19: 5h = 20, so h = 4. At exactly 4 hours the uncapped cost is 5(4) - 1 = $19, equal to the cap. For stays longer than 4 hours the uncapped formula exceeds $19, so the cap first reduces the bill beyond 4 hours. At 4 hours the customer pays $19 either way.
For 6 hours at the café: 5(6) - 1 = $29 uncapped, so the customer pays the $19 cap. The competitor charges 6(3.25) = $19.50. The café is cheaper by $0.50 for a 6-hour stay. The comparison would reverse for shorter stays, where the café's half-hour rounding and higher early rate make it relatively expensive; the cap only helps long visits.