NCDOT looks at the following metrics when analyzing traffic data:

$$ Crash Rate, which is defined to be the total crashes per $100$ million vehicle miles traveled $($VMT$).$

$$ Injury Rate, which is defined to be the total injuries per $100$ million VMT$.$

$$ Fatality Rate, which is defined to be the total fatalities per $100$ million VMT$.$

The following table contains the VMT data $($measured in hundreds of millions VMTs$)$ for Wake and Durham Counties $2020-2023$

$2020-2023$ VMT Data $(100$ Million VMT$)$

Year

$2023$

$2022$

$2021$

$2020$

Wake County

$126\mathrm{.}12119\mathrm{.}13113\mathrm{.}2899\mathrm{.}63$

Durham County

$39\mathrm{.}684037\mathrm{.}0934\mathrm{.}15$

$1000126\mathrm{.}12$

$0100$

Let U $=$

$119\mathrm{.}13$

$1.$

$00113\mathrm{.}280$

$199\mathrm{.}63$

$($b$)[5$ points$]$ What is the purpose of the matrix U$?$

$($c$)[5$ points$]$ Construct and compute a matrix product that produces the crash, injury, and fa$-$

tality rates for Durham County in the years $2020-2023.$

$($d$)[5$ points$]$ Consider again the following statement:

Based on the data, Durham County roads are overwhelmingly safer than Wake County roads.

Use your calculations in $($a$)-($c$)$ to explain why you agree or disagree with the statement above. Give a brief explanation as to whether or not your answer agrees with your work on Problem $2($b$).$

$000$

$($a$)[5$ points$]$ If possible, compute ATU. Round answers to two decimal places.