#### Answer

See the full explanation below.

#### Work Step by Step

(a)
Consider the provided statement “In $1954$, Roger Bannister of Britain cracked the $4-\text{minute}$ mark, setting the record for running a mile in 3 minutes, 59.4 seconds, or 239.4 seconds. In the half century then, the record has decreased by $0.3$ seconds per year.”
Thus, to express the record time for the mile run, the linear function representing the record time for a mile run will be of the form $M\left( x \right)=mx+b$.
Suppose that running x miles will consume equal time in total for a mile run after $1954$.
Therefore, the record time for setting the record for running a mile in 239.4 seconds after running x miles and a year which is decreased by $0.3$ second per year is:
$239.4-\left( 0.3 \right)x$.
Thus, by above given conditions
$M\left( x \right)=239.4-0.3x$
Hence, to express the record time for the mile run:
$M\left( x \right)=239.4-0.3x$.
(b)
Consider the provided statement “In $1954$, Roger Bannister of Britain cracked the $4-\text{minute}$ mark, setting the record for running a mile in 3 minutes, 59.4 seconds, or 239.4 seconds. In the half century then, the record has decreased by $0.3$ seconds per year.”
Thus, to calculate the year in which someone will run a $3-\text{minute}$, or $180-\text{second}$ mile use the linear function representing the record time for a mile run of the form $M\left( x \right)=239.4-0.3x$ from part (a).
Therefore, from the above given condition.
$\begin{align}
& M\left( x \right)=239.4-0.3x \\
& 180=239.4-0.3x \\
& 59.4=0.3x \\
& 198=x
\end{align}$
Hence, the year in which someone will run a $3-\text{minute}$ , or $180-\text{second}$ mile is $198\text{ years}$ after 1954, in 2152.