The stopping distance of an automobile is the distance...

The stopping distance of an automobile is the distance traveled during the driver's reaction time plus the distance traveled after the brakes are applied. In an experiment, these distances were measured (in feet) when the automobile was traveling at a speed of $x$ miles per hour on dry, level pavement, as shown in the bar graph. The distance traveled during the reaction time $R$ was $R=1.1 x$, and the braking distance $B$ was $B=0.0475 x^{2}-0.001 x+0.23 .$ a. Determine the polynomial that represents the total stopping distance $T$. b. Use the result of part (a) to estimate $T$ when $x=30, x=40,$ and $x=55.$ c. Use the bar graph to make a statement about the total stopping distance required at increasing speeds.

The stopping distance of an automobile is the distance traveled during the
driver's reaction time plus the distance traveled after the brakes are applied. In an experiment, these distances were measured (in feet) when the automobile was traveling at a speed of $x$ miles per hour on dry, level pavement, as shown in the bar graph. The distance traveled during the reaction time $R$ was $R=1.1 x$, and the braking distance $B$ was $B=0.0475 x^{2}-0.001 x+0.23 .$
a. Determine the polynomial that represents the total stopping distance $T$.
b. Use the result of part (a) to estimate $T$ when $x=30, x=40,$ and $x=55.$
c. Use the bar graph to make a statement about the total stopping distance required at increasing speeds.

Key Concepts

Graphical Analysis

Graphical analysis involves interpreting data presented in visual formats, such as bar graphs, to draw conclusions about trends and relationships. The bar graph in the problem provides a visual comparison of stopping distances at different speeds, which is used to support or validate the polynomial model's predictions.

Mathematical Modeling

Mathematical modeling is the use of mathematical expressions and equations to represent real-world scenarios. In this problem, mathematical modeling is applied to represent the total stopping distance of an automobile based on empirical formulas for reaction time and braking distance, reflecting how real-world phenomena can be analyzed quantitatively.

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients combined using operations of addition, subtraction, multiplication, and non-negative integer exponents. In the context of the problem, recognizing and manipulating polynomial expressions is essential for combining the components of stopping distance into a single mathematical model.

Combining Functions

Combining functions involves the addition, subtraction, or other operations between two or more functions to create a new function that captures a combined behavior. Here, the reaction distance and braking distance functions are combined, which is a critical step in formulating the total stopping distance as a polynomial.

Function Evaluation

Function evaluation is the process of substituting specific values into a function to determine the function's output at those points. This process is used to estimate the stopping distance for particular speeds by substituting the speed values into the polynomial model.

Transcript

00:01 So in this problem, they give us two different equations here.

00:03 R represents the reaction time, and b represents the breaking distance.

00:08 So in part a of this problem, they want us to come up with a polynomial to represent the total stopping distance.

00:14 Well, think about it.

00:15 You're going to need time to react, and then the time to break.

00:19 So to find the total time it's going to take to stop, we have to take the reaction time, r, and add it to the break time, which is b.

00:28 So we need to add the two polynomials 1 .1x plus 0 .0475 x squared minus 0 .001x plus 0 .23.

00:42 And now we're going to simplify, and i'm going to put this in standard form.

00:46 So i'm going to have my x squared term first, so 0 .0475 x squared.

00:52 Next, i'll combine the x terms.

00:54 So i have 1 .1x minus 0 .001x, minus 0 .001x.

00:58 Which will give us positive 1 .099x, and then we'll bring down the constant term 0 .23.

01:06 So now we have found the polynomial to represent the total stopping distance.

01:12 All right, so now let's move on to part b.

01:14 In part b, they want us to use this result in part a to estimate the value for t for different x values.

01:22 So first, we have when x is 30.

01:25 So if we need to find the total time it's going to take to stop, which is t, all we need to do is substitute 30 in place of x.

01:33 So when i do that, for the first one, we'll have 0 .0475 times 30 squared, plus 1 .099 times 30 plus 0 .23.

01:48 So now, if i were you, i would go to your calculator and just punch this in.

01:53 So you can have 0 .0475 times 30 squared.

01:57 And then you can add 1 .099 times 30 plus 0 .03, or sorry, 0 .23, and that will give us 75 .95.

02:12 So again, this will tell us our total stopping time.

02:18 Next, or stopping distance, i should say...

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