Problem Analysis:

Part (a):

• The motorist travels 90 km at 60 km/h and then for 1.5 hours at 80 km/h.
• Traffic police arrest motorists traveling over 80 km/h. We need to determine if the motorist exceeds this speed.

Part (b):

1. Calculate stopping distances using the formula:
   $d = \frac{s^2}{200} + \frac{s}{10}$
   for 46 km/h and 85 km/h.

2. Find the speed $s$ when the stopping distance $d$ is 35 meters.

Let me solve each part step-by-step. Defining the required functions and parameters

Function to calculate stopping distance

def stopping_distance(s): return (s**2)/200 + s/10

Values for stopping distances at 46 km/h and 85 km/h

speed_46 = 46 # km/h speed_85 = 85 # km/h distance_46 = stopping_distance(speed_46) distance_85 = stopping_distance(speed_85)

Function to find speed for a given stopping distance (solve for s when d = 35)

import sympy as sp

s = sp.Symbol('s', positive=True) # speed symbol d_eq = (s**2)/200 + s/10 - 35 # equation for d = 35 speed_for_35 = sp.solve(d_eq, s)

distance_46, distance_85, speed_for_35### Results:

1. The stopping distance for:

   • $46 \, \text{km/h}$: $15.18 \, \text{m}$
   • $85 \, \text{km/h}$: $44.63 \, \text{m}$

2. The speed $s$ at which the stopping distance is $35 \, \text{m}$ is approximately: $s \approx -10 + 10\sqrt{71} \, \text{km/h} \approx 74.2 \, \text{km/h}$

Next, I will calculate if the motorist is arrested based on his speeds. Checking if the motorist is arrested based on their speeds

First journey: 90 km at 60 km/h

speed_1 = 60 # in km/h distance_1 = 90 # in km time_1 = distance_1 / speed_1 # in hours

Second journey: 1.5 hours at 80 km/h

speed_2 = 80 # in km/h time_2 = 1.5 # in hours distance_2 = speed_2 * time_2

Max speed encountered by the motorist

max_speed = max(speed_1, speed_2) max_speed### Part (a) Result:

The maximum speed the motorist reached during the journey was 80 km/h, which is not above the speed limit.
Conclusion: The motorist will not be arrested.

Would you like a detailed explanation for any step?

Related Questions:

1. What is the derivation of the stopping distance formula $d = \frac{s^2}{200} + \frac{s}{10}$?
2. How can you generalize this problem for speeds in $\text{m/s}$ instead of $\text{km/h}$?
3. Why is the speed at $35 \, \text{m}$ calculated using symbolic solutions?
4. What would be the time saved by driving at 80 km/h for the entire journey instead of two segments?
5. Can the stopping distance formula be extended for wet or icy roads?

Tip: For practical problems like this, ensure unit consistency (e.g., $\text{m/s}$ vs. $\text{km/h}$).