A rental car costs 35 per day plus .10 per mile. a. What...

A rental car costs $$\$ 35$$ per day plus $$\$ .10$$ per mile. a. What is the cost, in dollars, to rent the car for 5 days if you drive a total of $M$ miles? b. What is the cost to rent the car for 5 days if you drive $M$ miles each day? c. What is the cost to rent the car for $D$ days if you drive a total of 85 miles? d. What is the cost to rent the car for $D$ days if you drive 85 miles each day? e. What would be the cost per person if three people share the cost of renting the car in Part $\mathrm{d}$ ?

A rental car costs $$\$ 35$$ per day plus $$\$ .10$$ per mile.
a. What is the cost, in dollars, to rent the car for 5 days if you drive a total of $M$ miles?
b. What is the cost to rent the car for 5 days if you drive $M$ miles each day?
c. What is the cost to rent the car for $D$ days if you drive a total of 85 miles?
d. What is the cost to rent the car for $D$ days if you drive 85 miles each day?
e. What would be the cost per person if three people share the cost of renting the car in Part $\mathrm{d}$ ?

Key Concepts

Linear Functions

A linear function represents a relationship where the dependent variable is a linear combination of independent variables, typically taking the form f(x) = mx + b, where m is the slope and b is the y-intercept. In the context of cost problems, the fixed rental fee and the per-mile charge combine to form such a linear relationship, allowing us to compute total cost based on the number of days and miles driven.

Fixed and Variable Costs

Fixed costs are expenses that do not change with the level of activity (e.g., the daily rental charge), while variable costs change with the level of usage (e.g., the per-mile charge). Recognizing both costs is key to modeling real-world scenarios in mathematics and helps in forming the overall cost function.

Translating Word Problems into Mathematical Expressions

This concept involves reading a descriptive scenario and identifying relevant quantities, variables, and their relationships. The ability to convert words into mathematical expressions or equations is essential for problem solving in algebra, particularly when determining total cost based on given parameters.

Total Cost Calculation

Total cost calculation combines the fixed cost component with the variable cost component into a single expression. This demonstrates how to sum different parts of a problem (e.g., cost per day and cost per mile), enabling a comprehensive understanding of the underlying financial model.

Cost Sharing

Cost sharing involves dividing the total cost equally among a number of parties. This concept not only requires calculating the overall cost but also applying division to determine how much each participant is responsible for, which is common in problems involving splitting expenses.

Transcript

00:01 So this problem is set up by telling us that at a certain car rental agency, a compact car rents for $30 a day and $10 a mile.

00:08 Part a asks us for what the price is of renting a car for three days and driving it for total of 280 miles.

00:17 So first we're going to start with finding the cost of the days alone.

00:21 So we know that it's going to be $30 for each day times a total of three days.

00:28 So we can do this because we know $1 or one day is $30.

00:34 So two days will be 30 times two, three days will be 30 times three, and so on.

00:40 So the day rental is going to be a total of $90.

00:45 Then we're going to calculate the price for the mileage that we're putting on the car.

00:51 So we know that it's 10 cents a mile and we're going to convert this to $2 so that everything is easier.

00:56 So we're going to say $0 .1, since that's equivalent to 10 cents per mile times our 280 miles driven.

01:08 And this is for the same reason as multiplying the $30 per day times our three days.

01:14 We're also multiplying our 10 cents per mile times 280 miles.

01:18 And that gives us a total of $28 for the mileage.

01:24 And then to calculate the total cost, we're just going.

01:28 To add these two costs together so it's going to be ninety dollars plus twenty eight dollars so our final cost is going to be one hundred and eighteen dollars and then part b asks us to basically create a model to represent this this function so we're just going to use the steps that we did in part a to come up with our model...

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