The Phone Plan Decision
Maya is choosing between two mobile phone plans. Plan A charges a flat $18 per month plus $0.10 per minute of calls. Plan B charges $30 per month with the first 200 minutes free, then $0.05 per minute beyond 200. Maya's monthly call minutes vary; last month she used 260 minutes, and she estimates her usage is between 150 and 350 minutes depending on the month. She wants a rule for which plan is cheaper at a given usage level m (in minutes).
1. Write cost functions A(m) and B(m) and determine the usage at which the two plans cost the same.
Read the worked answer
A(m) = 18 + 0.10m. For B: if m ≤ 200, B(m) = 30; if m > 200, B(m) = 30 + 0.05(m − 200). For m > 200, set 18 + 0.10m = 30 + 0.05(m − 200). Right side: 30 + 0.05m − 10 = 20 + 0.05m. So 18 + 0.10m = 20 + 0.05m → 0.05m = 2 → m = 40. But m = 40 is not in the region m > 200, so no crossover exists there. For m ≤ 200: 18 + 0.10m = 30 → 0.10m = 12 → m = 120. So they are equal at 120 minutes; below 120, Plan A is cheaper; above 120, Plan B is cheaper (check m = 260: A = 44, B = 33).
2. Maya says she will 'usually' use 260 minutes. Which plan should she pick, and how much does she save versus the other?
Read the worked answer
At m = 260: A = 18 + 0.10(260) = 18 + 26 = $44. B = 30 + 0.05(60) = 30 + 3 = $33. Plan B saves $11 per month. Since 260 > 120, Plan B is cheaper for all usage above 120 minutes, so the choice is robust across her 150–350 range.
3. Maya's usage could be as low as 150 minutes. Does the recommendation change anywhere in the 150–350 range? Justify.
Read the worked answer
For any m in [150, 350], m > 120, and we established Plan B is cheaper for all m > 120 (and equal at 120). Check endpoints: at 150, A = 33, B = 30 (B cheaper by $3); at 350, A = 53, B = 30 + 0.05(150) = 37.5 (B cheaper by $15.50). The recommendation does not change; Plan B is cheaper throughout.