A piecewise cost changes its daily rate after a stated number of days. To compare it with a flat fee, first check which rate applies when the costs become equal. A Desmos regression can find that break-even day. The lower daily rate later does not erase earlier charges, and “less than” excludes a tie.
Hints
- Hint 1
The unit rate is $6 for each of the first 100 days parked. If the student parks for days in that range, what is the pay-as-you-go total?
- Hint 2
Check the cost at day 100. Each day after that still adds $2, so can the total drop back below the permit price?
- Hint 3
The break-even day is when the two plans cost the same. The question asks when the permit costs less, and a student can park only a whole number of days.
Step-by-step
Find the break-even day
Step 1Write the cost for the first 100 days
Let be the whole number of days parked. For the first 100 days, each parked day costs $6, so the pay-as-you-go total is dollars for .
- Step 2
Check which pricing tier matters
Type in Desmos. It shows , already above the permit's $480. Each additional day adds $2, so the total never drops again. The first day the permit is cheaper must be in the $6-per-day tier.
- Step 3
Translate “cost less than”
The permit costs $480 once. For the permit to cost less than the pay-as-you-go plan in this tier, write:
First find the break-even point, where the two sides are equal. That marks the boundary, but it does not satisfy the strict inequality.
- Step 4
Find the equal-cost day
Type in Desmos. The asks Desmos to fit , the day count. Under PARAMETERS, it shows . At 80 days, both plans cost $480, so the permit is not cheaper yet.
- Step 5
Take the first whole day after the tie
A tie is not “less than,” so move one whole day past it. Type ; Desmos shows . That is still within the first 100 days, when each extra day raises the pay-as-you-go cost by $6. The minimum is 81 days. Grid in 81.