Question
Braking Distance The table below gives the results of an online calculator showing how far (in feet) a vehicle will travel while braking to a complete stop, given the initial velocity of the automobile.Braking Distance$$\begin{array}{c|c}\text { MPH } & \text { Distance (feet) } \\\hline 10 & 27 \\\hline 20 & 63 \\\hline 30 & 109 \\\hline 40 & 164 \\\hline 50 & 229 \\\hline 60 & 304 \\\hline 70 & 388 \\\hline 80 & 481 \\\hline 90 & 584 \\\hline\end{array}$$a. Find a quadratic model for the stopping distance.b. What other factors besides the initial speed would affect the stopping distance?
Step 1
We can use the form of a quadratic function, which is $f(x) = ax^2 + bx + c$. To find the coefficients $a$, $b$, and $c$, we can use three points from the table. Let's use the points (10, 27), (20, 63), and (30, 109). Show more…
Show all steps
Your feedback will help us improve your experience
Carson Merrill and 67 other Algebra educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
The braking distance required for a car to stop depends on numerous variables such as the speed of the car, the weight of the car, reaction time of the driver, and the coefficient of friction between the tires and the road. For a certain vehicle on one stretch of highway, the braking distances $d(s)$ (in $\mathrm{ft}$ ) are given for several different speeds $s$ (in mph). $$ \begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{s} & 30 & 35 & 40 & 45 & 50 \\ \hline \boldsymbol{d} \boldsymbol{(} \boldsymbol{s}) & 109 & 134 & 162 & 191 & 223 \\ \hline \boldsymbol{s} & 55 & 60 & 65 & 70 & 75 \\ \hline \boldsymbol{d}(\boldsymbol{s}) & 256 & 291 & 328 & 368 & 409 \\ \hline \end{array} $$ a. Use regression to find a quadratic function to model the data. b. Use the model from part (a) to predict the stopping distance for the car if it is traveling 62 mph before the brakes are applied. Round to the nearest foot. c. Suppose that the car is traveling 53 mph before the brakes are applied. If a deer is standing in the road at a distance of $245 \mathrm{ft}$ from the point where the brakes are applied, would the car hit the deer?
Polynomial and Rational Functions
Quadratic Functions and Applications
Stopping Distance When the driver of a vehicle observes an impediment, the total stopping distance involves both the reaction distance (the distance the vehicle travels while the driver moves his or her foot to the brake pedal) and the braking distance (the distance the vehicle travels once the brakes are applied). For a car traveling at a speed of $v$ miles per hour, the reaction distance $R,$ in feet, can be estimated by $R(v)=2.2 v$. Suppose that the braking distance $B,$ in feet, for a car is given by $B(v)=0.05 v^{2}+0.4 v-15$ (a) Find the stopping distance function $D(v)=R(v)+B(v)$ (b) Find the stopping distance if the car is traveling at a speed of $60 \mathrm{mph}$. (c) Interpret $D(60)$.
Functions and Their Graphs
Functions
During the testing of a new automobile braking system, the speeds $x$ (in miles per hour) and the stopping distances $y$ (in feet) were recorded in the table. $$\begin{array}{|c|c|} \hline \text { Speed, } x & \text { Stopping distance, } y \\ \hline 30 & 55 \\ \hline 40 & 105 \\ \hline 50 & 188 \\ \hline \end{array}$$ (a) Use the data to create a system of linear equations. Then find the least squares regression parabola for the data by solving the system. (b) Use a graphing utility to graph the parabola and the data in the same viewing window. (c) Use the model to estimate the stopping distance for a speed of 70 miles per hour.
Linear Systems and Matrices
Multivariable Linear Systems
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD