Question 98·Medium·Linear Functions
A taxi company charges a flat fee of $4 when a passenger enters the cab plus $2.50 for each mile traveled.
Let be the total cost, in dollars, of a ride that covers miles.
Which equation models this relationship?
For word problems that ask for an equation, first identify the fixed amount (the value that does not change with the variable) and the rate (the amount per unit of the variable). Represent the total as total = rate × variable + fixed amount, then carefully match each number from the problem to its correct place: the coefficient of the variable is the rate, and the constant term is the fixed fee. Quickly test by plugging in an easy value like 0 or 1 to see if the equation matches the described situation.
Hints
Separate fixed and changing parts
Ask yourself: What amount do you pay no matter how many miles you ride, and what amount depends on the number of miles ?
Think about a 0-mile ride
If the taxi somehow went 0 miles, what would the total cost be? That value should be the constant term in your equation (the part without ).
Match rate and constant to the equation
The number that multiplies should represent the dollars per mile, and the stand-alone number should represent the flat fee. Choose the option whose coefficient of matches the per-mile rate and whose constant term matches the flat fee.
Desmos Guide
Enter the four equations
Type each answer choice into Desmos as a separate line: C = 4m + 2.5, C = 2.5m + 4, C = 2m + 4.5, and C = 4m - 2.5. Use on the horizontal axis and on the vertical axis.
Check the flat fee using the y-intercept
For each line, look at the point where it crosses the vertical axis (where ). The correct equation should give a cost of $4 when , because the flat fee is $4 even if no miles are traveled.
Check the per-mile rate using a 1-mile ride
For each equation, either use a table (click the gear icon and add a table) or substitute into the expression in Desmos. The correct model will increase the cost by $2.50 when increases from 0 to 1 mile (from the flat fee cost to the cost after one mile). Choose the equation whose graph and values match both conditions.
Step-by-step Explanation
Identify the fixed cost and the per-mile cost
From the problem:
- The flat fee is $4. This is the amount you pay even if you travel 0 miles.
- The per-mile charge is $2.50 for each mile traveled.
So, the total cost has two parts: a fixed cost and a cost that depends on the number of miles .
Write expressions for each part of the total cost
Let be the total cost, and be the number of miles.
- The fixed cost is just $4.
- The variable cost (the part that changes with miles) is $2.50 for each mile, so it is represented by .
So in words, the total cost is:
total cost = flat fee + per-mile cost
or numerically,
total cost = $4 + .
Combine the parts into an equation
Now translate the verbal expression into an algebraic equation using for total cost:
- "Total cost" becomes .
- "4 + 2.5m" is the sum of the fixed fee and the per-mile charge.
So the equation that models the relationship is
This matches answer choice B.