Two ways to cross
Building a linear model from a description, then using it to answer three questions the description does not answer directly.
Scenario
A ferry company charges a fixed boarding fee plus a per-mile rate. A 4-mile crossing costs $11. A 10-mile crossing costs $20.
A rival company charges no boarding fee and $2.25 per mile on every route.
Answers in order, explanations withheld, and a report on where the chain broke.
- 1.1Systems of equationsStandard
What is the ferry company's per-mile rate?
- A$1.50
- B$2.00
- C$2.75
- D$1.10
Show the answer and reasoningHide the reasoning
$1.50
Two points on a line give its slope: the cost rises $9 across 6 miles, so the rate is 9 / 6 = $1.50 per mile. Nothing about the fixed fee is needed yet.
- 1.2Linear functionsStandard
Builds on: The $1.50 per-mile rate found in question 1.
What is the ferry company's fixed boarding fee?
- A$5.00
- B$3.00
- C$7.00
- D$4.50
Show the answer and reasoningHide the reasoning
$5.00
Substitute a known crossing into cost = fee + 1.50 x miles. Using the 4-mile crossing: 11 = fee + 6, so the fee is $5. The 10-mile crossing checks it: 5 + 15 = 20.
- 1.3Systems of equationsChallenge
Builds on: Both the rate from question 1 and the fee from question 2 — the full model.
At what crossing length do the two companies charge the same amount?
- AAbout 6.7 miles
- BAbout 5.0 miles
- CAbout 8.0 miles
- DAbout 3.3 miles
Show the answer and reasoningHide the reasoning
About 6.7 miles
Set the two models equal: 5 + 1.50m = 2.25m. Subtracting gives 5 = 0.75m, so m = 20/3, about 6.7 miles. Below that the rival is cheaper; above it the ferry is.
- 1.4Interpreting linear modelsChallenge
Builds on: The crossover point from question 3, which predicts which company wins before any arithmetic.
A traveller makes a 12-mile crossing. Which statement is true?
- AThe ferry is cheaper, by $4.00.
- BThe rival is cheaper, by $4.00.
- CThe two cost the same.
- DThe ferry is cheaper, by $2.00.
Show the answer and reasoningHide the reasoning
The ferry is cheaper, by $4.00.
The ferry charges 5 + 1.50(12) = $23; the rival charges 2.25(12) = $27. The ferry is $4 cheaper, which is consistent with 12 miles being past the 6.7-mile crossover found in question 3.