Mathematics - Choosing a Data Plan When Usage Is Unknown
A student is planning a mobile-data budget for a semester. Plan A charges KD 5 per month plus KD 0.300 per GB above 10 GB. Plan B charges KD 9 per month plus KD 0.100 per GB above 10 GB. The student uses between 10 and 40 GB per month, but the exact usage m is unknown. A classmate graphs both plans and claims Plan B is always cheaper because its per-GB rate is lower. The student has KD 60 for the four-month semester and wants a recommendation that is valid for all possible usages in the stated range, not just an average. Both plans include 10 GB; there are no other fees, and the student wants to avoid any month exceeding the total budget.
1. Write the monthly cost functions A(m) and B(m) for 10 ≤ m ≤ 40, and find the usage at which the two plans cost the same.
Read the worked answer
For 10 ≤ m ≤ 40, the chargeable usage above the included 10 GB is (m − 10).
- Plan A: A(m) = 5 + 0.300(m − 10)
- Plan B: B(m) = 9 + 0.100(m − 10)
Set them equal:
5 + 0.300(m − 10) = 9 + 0.100(m − 10)
5 + 0.300m − 3 = 9 + 0.100m − 1
2 + 0.300m = 8 + 0.100m
0.200m = 6, so m = 30 GB.
At m = 30, A = 5 + 0.300(20) = 11 KD and B = 9 + 0.100(20) = 11 KD. The intersection is real and inside the stated range, so the classmate's 'always cheaper' claim already fails on one side of 30 GB.
2. The classmate's claim is false for part of the range. State the correct condition and evaluate the four-month budget constraint.
Read the worked answer
Compare the slopes: A rises by 0.300 KD per GB, B by 0.100 KD per GB, so B is cheaper only once its lower slope has overcome its higher fixed fee. Because they meet at 30 GB:
- For 10 ≤ m < 30, A(m) < B(m); Plan A is cheaper.
- At m = 30, A = B = 11 KD.
- For 30 < m ≤ 40, B(m) < A(m); Plan B is cheaper.
Budget check: the worst case for Plan A in the range is m = 40, giving A = 5 + 0.300(30) = 14 KD per month, or 56 KD over four months. The worst case for Plan B is B = 9 + 0.100(30) = 12 KD per month, or 48 KD over four months. Both fit under KD 60, so the budget does not force the choice. The correct recommendation depends on expected usage: A below 30 GB, B above 30 GB. The classmate looked only at the per-GB rate and ignored the fixed fee, which is the error.
3. If usage is unknown and every value from 10 to 40 GB is equally likely, which plan has the lower expected cost? Does that change your recommendation?
Read the worked answer
For a uniform distribution on [10, 40], the mean usage is the midpoint, 25 GB. Because both cost functions are linear, the expected cost equals the cost at the mean:
- E[A] = 5 + 0.300(25 − 10) = 5 + 4.50 = 9.50 KD per month.
- E[B] = 9 + 0.100(25 − 10) = 9 + 1.50 = 10.50 KD per month.
Plan A has the lower expected cost by 1 KD per month. But the recommendation should not ignore risk: Plan A's worst case is 14 KD per month, while Plan B's worst case is 12 KD. If the student can tolerate the possible 14 KD month, A is the better expected-value choice; if the student wants the smaller worst-case bill, B is safer. The judgement is that 'lower expected cost' and 'lower maximum cost' are different criteria, and the problem asks for a recommendation valid across the range, so the answer should state the criterion rather than only name a plan.