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.

\(\displaystyle f{{\left({x}\right)}}=-{0.554}{x}^{2}+{35.5}{x}-{514}\)

\(\displaystyle f{{\left({28}\right)}}=-{0.554}{\left({28}\right)}^{2}+{35.5}{\left({28}\right)}-{514}\)

\(\displaystyle=-{434.336}+{994}-{514}\)

\(\displaystyle f{{\left({28}\right)}}={45.67}\)

\(\displaystyle f{{\left({30}\right)}}=-{0.554}{\left({30}\right)}^{2}+{35.5}{\left({30}\right)}-{514}\)

\(\displaystyle=-{498.6}+{1065}-{514}\)

\(\displaystyle f{{\left({30}\right)}}={52.4}\)

\(\displaystyle f{{\left({32}\right)}}=-{0.554}{\left({32}\right)}^{2}+{35.5}{\left({32}\right)}-{514}\)

\(\displaystyle=-{567.296}+{1136}-{514}\)

\(\displaystyle f{{\left({32}\right)}}={54.704}\)

\(\displaystyle f{{\left({34}\right)}}=-{0.554}{\left({34}\right)}{2}+{35.5}{\left({34}\right)}-{514}\)

\(\displaystyle=-{640.424}+{1207}-{514}\)

\(\displaystyle f{{\left({34}\right)}}={52.58}\)

\(\displaystyle f{{\left({36}\right)}}=-{0.554}{\left({36}\right)}^{2}+{35.5}{\left({36}\right)}-{514}\)

\(\displaystyle=-{717.984}+{1278}-{514}\)

\(\displaystyle f{{\left({36}\right)}}={46.02}\)

(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.