Question

# An automobile tire manufacturer collected the data in the table relating tire pressure x (in pounds per square inch) and mileage (in thousands of mile

Modeling data distributions
An automobile tire manufacturer collected the data in the table relating tire pressure x​ (in pounds per square​ inch) and mileage​ (in thousands of​ miles). A mathematical model for the data is given by
$$\displaystyle​ f{{\left({x}\right)}}=-{0.554}{x}^{2}+{35.5}{x}-{514}.$$
$$\begin{array}{|c|c|} \hline x & Mileage \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}$$
​(A) Complete the table below.
$$\begin{array}{|c|c|} \hline x & Mileage & f(x) \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}$$
​(Round to one decimal place as​ needed.)
$$A. 20602060xf(x)$$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,45), (30,51), (32,56), (34,50), and (36,46). A parabola opens downward and passes through the points (28,45.7), (30,52.4), (32,54.7), (34,52.6), and (36,46.0). All points are approximate.
$$B. 20602060xf(x)$$
Acoordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2.
Data points are plotted at (43,30), (45,36), (47,41), (49,35), and (51,31). A parabola opens downward and passes through the points (43,30.7), (45,37.4), (47,39.7), (49,37.6), and (51,31). All points are approximate.
$$C. 20602060xf(x)$$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,45), (45,51), (47,56), (49,50), and (51,46). A parabola opens downward and passes through the points (43,45.7), (45,52.4), (47,54.7), (49,52.6), and (51,46.0). All points are approximate.
$$D.20602060xf(x)$$
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,30), (30,36), (32,41), (34,35), and (36,31). A parabola opens downward and passes through the points (28,30.7), (30,37.4), (32,39.7), (34,37.6), and (36,31). All points are approximate.
​(C) Use the modeling function​ f(x) to estimate the mileage for a tire pressure of 29
$$\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ and for 35
$$\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$
The mileage for the tire pressure $$\displaystyle{29}\frac{{{l}{b}{s}}}{{{s}{q}}}\in.$$ is
The mileage for the tire pressure $$\displaystyle{35}\frac{{{l}{b}{s}}}{{{s}{q}}}$$ in. is
(Round to two decimal places as​ needed.)
(D) Write a brief description of the relationship between tire pressure and mileage.
A. As tire pressure​ increases, mileage decreases to a minimum at a certain tire​ pressure, then begins to increase.
B. As tire pressure​ increases, mileage decreases.
C. As tire pressure​ increases, mileage increases to a maximum at a certain tire​ pressure, then begins to decrease.
D. As tire pressure​ increases, mileage increases.

2021-03-12
Given that
$$\displaystyle f{{\left({x}\right)}}=-{0.554}{x}^{2}+{35.5}{x}-{514}$$
$$\displaystyle f{{\left({28}\right)}}=-{0.554}{\left({28}\right)}^{2}+{35.5}{\left({28}\right)}-{514}$$
$$\displaystyle=-{434.336}+{994}-{514}$$
$$\displaystyle f{{\left({28}\right)}}={45.67}$$
$$\displaystyle f{{\left({30}\right)}}=-{0.554}{\left({30}\right)}^{2}+{35.5}{\left({30}\right)}-{514}$$
$$\displaystyle=-{498.6}+{1065}-{514}$$
$$\displaystyle f{{\left({30}\right)}}={52.4}$$
$$\displaystyle f{{\left({32}\right)}}=-{0.554}{\left({32}\right)}^{2}+{35.5}{\left({32}\right)}-{514}$$
$$\displaystyle=-{567.296}+{1136}-{514}$$
$$\displaystyle f{{\left({32}\right)}}={54.704}$$
$$\displaystyle f{{\left({34}\right)}}=-{0.554}{\left({34}\right)}{2}+{35.5}{\left({34}\right)}-{514}$$
$$\displaystyle=-{640.424}+{1207}-{514}$$
$$\displaystyle f{{\left({34}\right)}}={52.58}$$
$$\displaystyle f{{\left({36}\right)}}=-{0.554}{\left({36}\right)}^{2}+{35.5}{\left({36}\right)}-{514}$$
$$\displaystyle=-{717.984}+{1278}-{514}$$
$$\displaystyle f{{\left({36}\right)}}={46.02}$$
(A) The completed table is
$$\begin{array}{|c|c|} \hline x & Mileage & f(x) \\ \hline 28 & 45 & 45.67 \\ \hline 30 & 51 & 52.4\\ \hline 32 & 56 & 54.7\\ \hline 34 & 50 & 52.58\\ \hline 36 & 46 & 46.02\\ \hline \end{array}$$
(C) Use the modeling function​ f(x) to estimate the mileage for a tire pressure of $$\displaystyle{29}\frac{{​{l}{b}{s}}}{{{s}{q}}}$$ in. and for $$\displaystyle{35}​\frac{{{l}{b}{s}}}{{{s}{q}}}$$ in.:
The mileage for the tire pressure $$\displaystyle{29}​\frac{{{l}{b}{s}}}{{{s}{q}}}$$ in. is
$$\displaystyle f{{\left({29}\right)}}=-{0.554}{\left({29}\right)}^{2}+{35.5}{\left({29}\right)}-{514}$$
$$\displaystyle=-{465.914}+{1029.5}-{514}$$
$$\displaystyle f{{\left({29}\right)}}={52.59}$$
The mileage for the tire pressure $$\displaystyle{35}​\frac{{{l}{b}{s}}}{{{s}{q}}}$$ in. is
$$\displaystyle f{{\left({35}\right)}}=-{0.554}{\left({35}\right)}^{2}+{35.5}{\left({35}\right)}-{514}$$
$$\displaystyle=-{678.65}+{1242.5}-{514}$$
$$\displaystyle f{{\left({35}\right)}}={49.85}$$
​(D) To write a brief description of the relationship between tire pressure and mileage:
From the table it is clear that, as tire pressure increases mileage increases for 28, 30 and attains its maximum value at 32, and began to decrease for 34,36.
Hence, as tire pressure​ increases, mileage increases to a maximum at a certain tire​ pressure, then begins to decrease.