Choosing Between Two Data Plans
A family is choosing between two monthly phone plans for a middle-school student. Plan A costs $30 per month and includes 4 GB of data; each additional gigabyte costs $6. Plan B costs $50 per month and includes 10 GB; each additional gigabyte costs $4. The family expects the student to use somewhere between 3 and 14 GB per month, and data is billed in whole gigabytes, but the family wants a continuous model to compare plans fairly. They also received a promotional flyer claiming that 'Plan B is always the better deal once you need more than 4 GB.' Before switching, the family wants to know exactly where the costs cross, how much they would save at their most likely usage of 9 GB, and whether the flyer's claim holds across the entire range they are considering.
1. Write a cost function for each plan over the range 3 to 14 GB, then solve for the usage where the two plans cost the same. Explain what that value means and whether there is more than one such point.
Read the worked answer
Let x be gigabytes used. Plan A: for 0 <= x <= 4, A(x) = 30; for x > 4, A(x) = 30 + 6(x - 4) = 6x + 6. Plan B: for 0 <= x <= 10, B(x) = 50; for x > 10, B(x) = 50 + 4(x - 10) = 4x + 10.
Check each region separately. On 4 < x <= 10, set 6x + 6 = 50, so 6x = 44 and x = 44/6 = 7 1/3. This is inside the region: at 7 GB, Plan A costs $48 and Plan B costs $50; at 8 GB, Plan A costs $54 and Plan B costs $50. So the crossing is real.
On x > 10, set 6x + 6 = 4x + 10, giving 2x = 4 and x = 2, which is not in that region, so no second crossing exists. At 10 GB, Plan A is $66 and Plan B is $50, and Plan B stays cheaper beyond that.
Meaning: below 7 1/3 GB, Plan A costs less; above 7 1/3 GB, Plan B costs less. There is exactly one crossing in the family's range.
2. At the family's most likely usage of 9 GB, which plan is cheaper and by how much per year? Show the substitution and explain why the answer is consistent with your crossing point.
Read the worked answer
At x = 9, use the x > 4 rule for Plan A and the x <= 10 rule for Plan B.
Plan A: A(9) = 30 + 6(9 - 4) = 30 + 30 = $60. Plan B: B(9) = $50 because 9 GB is within the 10 GB allowance.
Plan B saves $60 - $50 = $10 per month. Over 12 months that is 12($10) = $120 per year.
This is consistent with the crossing at 7 1/3 GB: 9 GB is above the crossing, so the plan that is cheaper above the crossing, Plan B, should win. The $10 difference also matches the linear expressions: at 9 GB, A = 6(9) + 6 = 60 and B = 50.
3. Evaluate the flyer's claim that Plan B is always better once usage exceeds 4 GB. Use at least two specific usage amounts and state the correct rule.
Read the worked answer
The claim is false. The crossing at 7 1/3 GB shows the plans are equal there, and Plan A is cheaper between 4 GB and 7 1/3 GB.
Test 5 GB: Plan A = 30 + 6(1) = $36; Plan B = $50. Plan A saves $14. Test 6 GB: Plan A = 30 + 6(2) = $42; Plan B = $50. Plan A saves $8. Test 7 GB: Plan A = $48; Plan B = $50. Plan A still saves $2.
At 8 GB the result reverses: Plan A = $54 and Plan B = $50. At 14 GB, Plan A = 6(14) + 6 = $90 and Plan B = 4(14) + 10 = $66, so Plan B saves $24.
Correct rule: Plan A is cheaper for usage below 7 1/3 GB, the plans tie at 7 1/3 GB, and Plan B is cheaper above 7 1/3 GB. The flyer's threshold of 4 GB is the point where Plan A's overage charges begin, not the point where Plan B becomes cheaper.