Phone Plan Break-Even with Piecewise Overage
A family is choosing between two monthly phone plans. Plan A costs $32 and includes 4 GB of data; each additional GB costs $8. Plan B costs $50 and includes 10 GB; each additional GB costs $4. Both plans charge only for whole additional gigabytes, and the family’s monthly usage is a whole number of gigabytes. They expect to use about 9 GB per month but want a plan that keeps the monthly bill under $65. They also want to know the smallest usage at which Plan B becomes cheaper than Plan A, and how much they would save at their expected usage.
1. Write piecewise cost functions A(g) and B(g) for g whole gigabytes of monthly usage. State the domain restriction implied by the situation.
Read the worked answer
For Plan A, the first 4 GB are included, so
$$A(g)=32 \quad \text{if } 0\le g\le 4,$$
and each GB above 4 adds $8:
$$A(g)=32+8(g-4)=8g \quad \text{if } g>4.$$
For Plan B, the first 10 GB are included:
$$B(g)=50 \quad \text{if } 0\le g\le 10,$$
and
$$B(g)=50+4(g-10)=4g+10 \quad \text{if } g>10.$$
Because usage is measured in whole gigabytes, g must be a nonnegative integer.
2. At 9 GB of usage, which plan is cheaper, and by how much?
Read the worked answer
Use the correct piece for each plan.
Plan A: $g=9>4$, so $A(9)=8(9)=72$ dollars.
Plan B: $g=9\le 10$, so $B(9)=50$ dollars.
Plan B is cheaper by $72-50=22$ dollars.
3. Under the $65 monthly budget, which plan allows more whole gigabytes, and what is the maximum usage for that plan?
Read the worked answer
For Plan A with $g>4$, $A(g)=8g\le 65$, so $g\le 8.125$. The largest whole number is 8 GB, costing $64.
For Plan B above 10 GB, $B(g)=4g+10\le 65$, so $4g\le 55$ and $g\le 13.75$. The largest whole number is 13 GB, costing $4(13)+10=62$ dollars.
Plan B allows more data: 13 GB versus 8 GB, while staying under $65.