Piecewise Delivery Fee and a Double-Counted Base
A food-delivery app charges $4.50 for the first 2 miles and $1.60 for each additional mile or part of a mile. A student writes f(m) = 4.50 + 1.60m for m ≥ 0, computes f(6) = 4.50 + 9.60 = $14.10, and submits it. The app's own example says a 6-mile delivery costs $10.90. The student checks her arithmetic twice, cannot find an error, and concludes that the app's example must be wrong. Her table of values shows f(0) = 4.50, f(2) = 7.70, f(3) = 9.30, and f(6) = 14.10. The tutor must identify the modeling error rather than an arithmetic one, decide how the 'or part of a mile' wording affects the rule, and help her decide whether her linear formula is ever correct.
1. Write the correct rule for the app's charge, including the effect of 'or part of a mile,' and evaluate it for 6 miles, 2.5 miles, and 2 miles.
Read the worked answer
The $4.50 covers the first 2 miles; the per-mile charge applies only to miles beyond 2. Let k = number of additional (possibly partial) miles = ceil(m - 2) for m > 2. Then:
- f(m) = 4.50 for 0 < m ≤ 2
- f(m) = 4.50 + 1.60·ceil(m - 2) for m > 2
Evaluations:
- 6 miles: 6 - 2 = 4 additional miles, so f(6) = 4.50 + 1.60(4) = $10.90, matching the app.
- 2.5 miles: 0.5 additional miles rounds up to 1, so f(2.5) = 4.50 + 1.60(1) = $6.10.
- 2 miles: no additional mileage, so f(2) = $4.50.
The student's formula charges 1.60 for all m miles, which charges again for the two miles already paid for in the base fee. The base fee is a bundled charge for the first 2 miles, not a fixed fee added on top of per-mile pricing from mile zero.
2. Explain the source of the student's error and quantify how far off her formula is for distances above 2 miles.
Read the worked answer
The error is interpretive: she read 'for the first 2 miles' as a flat fee that is charged in addition to a per-mile rate covering every mile, rather than as a fee that includes the first 2 miles. Her formula and the correct one differ by a constant for any m > 2 (ignoring the rounding rule):
(4.50 + 1.60m) - (4.50 + 1.60(m - 2)) = 1.60(2) = $3.20.
So she overcharges every delivery longer than 2 miles by exactly the price of the two miles already covered, and her table confirms it: at m = 3 she gets 9.30 instead of 6.10+... i.e., 4.50 + 1.60(1) = 6.10, a $3.20 gap; at m = 6 she gets 14.10 instead of 10.90, again $3.20. With the 'part of a mile' rule, the gap varies slightly at fractional distances but is about $3.20.
A good diagnostic habit is to test small, interpretable inputs: at m = 2 exactly, the app charges $4.50, but her formula gives $7.70 - the two rules fail to agree even on the boundary case, which shows the formula cannot be right for any m > 0.
3. The student asks whether her formula is ever correct. Explain when the two rules agree, and describe how the 'part of a mile' wording changes the graph of the correct function.
Read the worked answer
Set 4.50 + 1.60m = 4.50 + 1.60(m - 2) and the equation reduces to 0 = -3.20, which is false; the underlying linear pieces are parallel (same slope, different intercepts), so they never meet. Her formula agrees with the correct one only at m = 0, where both give $4.50 - and 0 miles is not a delivery the app handles. So her rule is not correct at any chargeable distance.
Rounding 'or part of a mile' makes the correct function a step function: at each whole-mile boundary above 2 the charge jumps by $1.60, and the graph consists of horizontal segments rather than a slanted line. For example, f(2.01) = 4.50 + 1.60 = 6.10, the same as f(3). If the app charged proportionally for partial miles, the second piece would be a line with slope 1.60, continuous at m = 2; with rounding, it is discontinuous at 2 and at each subsequent mile.
The choice between a linear and a step model is a modeling decision driven by the billing policy, not a matter of which looks simpler. A correct answer must state the domain of each piece and identify the rounding convention.