Car Rentals The cost C, in dollars, of a one-day car rental...

Car Rentals The cost $C,$ in dollars, of a one-day car rental is modeled by the function $C(x)=0.35 x+45,$ where $x$ is the number of miles driven. (a) What is the cost if you drive $x=40$ miles? (b) If the cost of renting the moving truck is $\$ 108$, how many miles did you drive? (c) Suppose that you want the cost to be no more than $\$ 150 .$ What is the maximum number of miles that you can drive? (d) What is the implied domain of $C ?$ (e) Interpret the slope. (f) Interpret the $y$ -intercept.

Car Rentals The cost $C,$ in dollars, of a one-day car rental is modeled by the function $C(x)=0.35 x+45,$ where $x$ is the number of miles driven.
(a) What is the cost if you drive $x=40$ miles?
(b) If the cost of renting the moving truck is $\$ 108$, how many miles did you drive?
(c) Suppose that you want the cost to be no more than $\$ 150 .$ What is the maximum number of miles that you can drive?
(d) What is the implied domain of $C ?$
(e) Interpret the slope.
(f) Interpret the $y$ -intercept.

Key Concepts

Domain of a Function

The domain of a function is the set of all possible input values for which the function is defined. Understanding the domain ensures that the applications of the function are realistic, especially in contexts where certain inputs may not be feasible.

Linear Function

A linear function represents a relationship where the output changes at a constant rate with respect to the input, typically written as y = mx + b. This concept is essential in mathematical modeling and real-world applications, as it describes how one quantity varies directly in relation to another.

Function Evaluation

Function evaluation involves substituting a specific input value into a function to compute its output. This operational skill is vital for applying mathematical models to determine specific outcomes in various scenarios.

Solving Linear Equations

This concept entails setting a linear function equal to a given value and solving for the variable. It is a fundamental algebraic technique used to determine unknown quantities based on known outputs.

Solving Linear Inequalities

Solving linear inequalities involves finding all input values that satisfy an inequality involving a linear function. This concept is particularly important in determining feasible ranges of values where certain constraints, like cost limits, are met.

Slope and y-Intercept Interpretation

The slope of a linear function represents the rate at which the output changes with respect to the input, indicating how much the dependent variable will increase or decrease as the independent variable changes. The y-intercept is the value of the function when the input is zero, often signifying a baseline or fixed initial amount in practical models.

Transcript

00:01 In this problem, we are interpreting linear equations using function notation.

00:06 So we have a function for a rental car.

00:09 The rental car costs $45 to rent for the day plus 35 cents for the number of miles that you drive.

00:17 So let's find the cost of renting the car if you are going to drive 40 miles.

00:23 So that would be the cost of driving 40 miles would be equal to 0 .35 times.

00:34 Times 40 plus 45 and that means your rental costs would be equal to $59 even so it would cost $59 to rent this car and drive for 40 miles.

01:11 What if you had a budget of no more than $150 so we would want our cost 150 to be less than or equal to 0 .35x plus 45 so let's solve that by subtracting 45 from both sides.

01:35 We get 105 equals 0 .35x.

01:41 We'll deal with the less than or equal to in a minute.

01:43 Let's divide both sides by 0 .35.

01:57 So it looks like that in order to pay $150 or less, we could drive less than or equal to.

02:15 Let me turn that around.

02:25 The number of miles that we could drive would be less than or equal to 300.

02:33 So if we drove 300 miles, it would be exactly $150 our budget, and we could drive less and pay less if we wanted to.

02:43 Okay, let's say that our bill was $108.

02:48 How many miles did we drive? this was actually part b, and this was part.

02:53 C.

02:54 So cost $108, how many miles did we drive? so $108 equals 0 .35x plus 45.

03:04 Let's subtract the 45 from each side...