Comparing Mobile Phone Plans
Mia is comparing two mobile phone plans. Plan A charges a flat rate of $30 per month and includes 5 GB of data; extra data costs $10 per GB. Plan B charges $20 per month and includes 2 GB of data; extra data costs $15 per GB. Mia estimates she will use between 3 GB and 7 GB of data per month. She wants to choose the plan that minimizes her monthly cost. However, she is unsure of her exact data usage and wants to know for what usage each plan is cheaper. Both plans charge in whole gigabytes for extra data (no partial gigabytes).
1. Write a piecewise function for the monthly cost of each plan as a function of data usage \( x \) GB, where \( x \) is a non-negative integer. Then determine the range of usage for which Plan A is cheaper than Plan B.
Read the worked answer
For Plan A: if ( 0 \le x \le 5 ), cost ( C_A = 30 ). If ( x > 5 ), cost ( C_A = 30 + 10(x - 5) = 10x - 20 ). For Plan B: if ( 0 \le x \le 2 ), cost ( C_B = 20 ). If ( x > 2 ), cost ( C_B = 20 + 15(x - 2) = 15x - 10 ). We want ( C_A < C_B ). Case 1: ( 0 \le x \le 2 ): ( C_A = 30 ), ( C_B = 20 ), so ( 30 < 20 ) false. Case 2: ( 2 < x \le 5 ): ( C_A = 30 ), ( C_B = 15x - 10 ). Solve ( 30 < 15x - 10 \Rightarrow 40 < 15x \Rightarrow x > 40/15 \approx 2.67 ). Since ( x ) integer, ( x \ge 3 ). So for ( x = 3,4,5 ), Plan A is cheaper. Case 3: ( x > 5 ): ( C_A = 10x - 20 ), ( C_B = 15x - 10 ). Solve ( 10x - 20 < 15x - 10 \Rightarrow -10 < 5x \Rightarrow x > -2 ), which is always true for ( x > 5 ). So for ( x > 5 ), Plan A is cheaper. Thus Plan A is cheaper for ( x \ge 3 ) GB. At ( x = 2 ), Plan B is cheaper; at ( x = 3 ), Plan A costs $30, Plan B costs $35, so Plan A cheaper. So Plan A is cheaper when usage is 3 GB or more.
2. Mia thinks she will use about 4 GB per month. Based on expected cost, which plan should she choose? What if she uses 6 GB? Show calculations.
Read the worked answer
For 4 GB: Plan A: since 4 ≤ 5, cost = $30. Plan B: since 4 > 2, cost = 20 + 15(4-2) = 20 + 30 = $50. So Plan A is cheaper by $20. For 6 GB: Plan A: 6 > 5, cost = 30 + 10(6-5) = $40. Plan B: 6 > 2, cost = 20 + 15(6-2) = 20 + 60 = $80. Plan A cheaper by $40. Thus for both 4 GB and 6 GB, Plan A is cheaper.