A tiered rate changes after a stated number of miles, so split the ride into miles before and after that cutoff. First find the most Kai can spend while keeping the required credit. Then use a Desmos regression to find where the fare reaches that limit. For a greatest whole number, stay below a fractional cutoff rather than rounding up.
Hints
- Hint 1
Kai must keep some of his credit. Subtract the required remaining credit from his starting balance to find the most the ride can cost.
- Hint 2
The flat fee is charged once. The first 15 miles all cost $1.20 per mile; if the ride is miles long, how many miles get the later rate?
- Hint 3
Set the fare equal to the most Kai can spend to find the cutoff. If Desmos gives a fractional number of miles, choose the greatest whole number below it.
Step-by-step
Model the two rates and find the fare cutoff
Step 1Find Kai’s fare limit
Kai starts with $30 and must keep at least $5, so the ride can cost no more than dollars.
- Step 2
Price the first 15 miles
Type in Desmos. It shows : the $2 flat fee, paid once, plus all 15 miles at the first rate.
- Step 3
Write the fare after 15 miles
Let be the fare in dollars for total miles, with . The first 15 miles still cost each, but only miles cost each. Type in Desmos. Only miles beyond 15 get the later rate. Kai’s fare limit means .
- Step 4
Find where the fare reaches the limit
The cutoff is where the fare equals . Type : Desmos uses as the unknown, and tells it to find the matching value. Under PARAMETERS, it shows . Each added mile raises the fare, so the allowed distances are at or below miles.
- Step 5
Choose the last affordable whole mile
Type and . Desmos shows and . The first fare fits the limit; the second does not. So rounding up would leave Kai with too little credit. The greatest whole distance is miles. Choice C.