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# Force and equilibrium: Hills, inclement weather, and cars

## Problem

Steep hills cause significant traffic safety problems when roads become wet or icy. The county commission office for a small town is attempting to identify the most dangerous hills in their city. They identified four roads on hills that seem to pose the most danger to drivers. Since the roads have different texture and are made of different materials, they all offer different amounts of traction between car tires and the road. The hills were tested under both icy and wet road conditions to determine the coefficients of static and kinetic friction between the road and an average car tire.
The results of the research are summarized in the table below:
Table 1. Coefficients of friction for each street at specific angles.
Hill locationHill angle${\text{μ}}_{s}$(icy)${\text{μ}}_{s}$(wet)${\text{μ}}_{k}$(icy)${\text{μ}}_{k}$(wet)
Bridger St${29}^{\circ }$0.530.610.210.46
Pacific Ave${33}^{\circ }$0.610.750.220.55
Austin Dr${24}^{\circ }$0.460.670.230.51
Western Ave${21}^{\circ }$0.370.410.200.32
Trigonometric values for various angles are given below:
Table 2. Angles and their corresponding trigonometric values.
Anglesinθcosθtanθ
${21}^{\circ }$0.3580.9340.384
${24}^{\circ }$0.4070.9140.445
${29}^{\circ }$0.4850.8750.554
${33}^{\circ }$0.5450.8390.659
Under icy conditions, which hill could a car park on without slipping?