Gas mileage depends in part on the speed of the vehicle. The...

Gas mileage depends in part on the speed of the vehicle. The gas mileage $m$ (in mpg) of a certain subcompact car is given by $m=-0.04 x^{2}+3.6 x-49,$ where $x$ is the speed of the car (in mph). (See Example 1) a. Use the model to approximate the gas mileage for a car traveling 35 mph. b. Use the model to approximate the gas mileage for a car traveling 50 mph. c. Use the model to approximate the gas mileage for a car traveling 75 mph.

Gas mileage depends in part on the speed of the vehicle. The gas mileage $m$ (in mpg) of a certain subcompact car is given by $m=-0.04 x^{2}+3.6 x-49,$ where $x$ is the speed of the car (in mph). (See Example 1)
a. Use the model to approximate the gas mileage for a car traveling 35 mph.
b. Use the model to approximate the gas mileage for a car traveling 50 mph.
c. Use the model to approximate the gas mileage for a car traveling 75 mph.

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Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form ax² + bx + c. They are widely used to model various phenomena, including optimization problems, trajectories, and in this context, fuel efficiency variations with speed. The behavior of the graph of a quadratic function, including its vertex and concavity, provides important insights into the maximum or minimum values attained by the function.

Function Evaluation

Function evaluation involves substituting a specific input value into a function to compute its output. This concept is fundamental in understanding how models work, as it allows one to predict the value of a quantity under specific conditions. In scenarios like testing gas mileage at various speeds, the function evaluation helps in estimating the efficiency of the vehicle for different driving speeds.

Mathematical Modeling

Mathematical modeling is the process of creating abstract representations of real-world situations using mathematical expressions or functions. Such models help in understanding, analyzing, and making predictions about real phenomena. In this context, the provided quadratic model approximates the gas mileage of a car at various speeds, offering insight into the relationship between speed and efficiency.

Transcript

00:01 So in this problem, we're given this equation.

00:02 M equals negative 0 .04x squared plus 3 .6x minus 49, and we're told that it's going to represent the gas mileage, which is m, based off the speed, which is x.

00:14 So let's go down to part a in this problem.

00:17 What we want to do is to use this model to approximate the gas mileage for a car that's traveling 35 miles per hour.

00:24 Well, that would mean the speed.

00:25 And don't forget, x represents the speed.

00:27 So in this case, x is going to be equal to 35.

00:32 So from here, all we're going to do is replace x with 35 in our equation, which means we'll have m equals negative 0 .04 times 35 squared plus 3 .6 times 35 minus 49.

00:49 Now, the nice thing from here is you can actually type this into your calculator all at once.

00:53 So if you do negative 0 .04, and then in parentheses, 309.

00:58 25, close preemphasy squared, plus 3 .6 times 35 minus 49, and press enter, now you'll have the approximate mileage.

01:08 So what you'll find is m will equal to 28, meaning it'll be approximately 28 miles per gallon.

01:15 So if we wanted to put units here, we can put miles per gallon.

01:21 Or actually, in the book, they're saying mpg.

01:24 So we'll just put that mpj...