You're on a roll — a) To determine if the driver will be arrested, we need to check their speed. Police arrest drivers traveling above $80 \text{ km/hr}$. The speed for the first part of the journey was $60 \text{ km/hr}$. The speed for the second part of the journey was $80 \text{ km/hr}$. Since neither speed is above $80 \text{ km/hr}$ ($60 \text{ km/hr} \ngtr 80 \text{ km/hr}$ and $80 \text{ km/hr} \ngtr 80 \text{ km/hr}$), the driver will not be arrested. The driver will not be arrested. b) We use the stopping distance formula $d = \frac{s^2}{200} + \frac{s}{10}$. To find the stopping distance for $s = 46 \text{ km/hr}$: Step 1: Substitute $s = 46$ into the formula. $$d = \frac{46^2}{200} + \frac{46}{10}$$ Step 2: Calculate $46^2$. $$d = \frac{2116}{200} + \frac{46}{10}$$ Step 3: Perform the divisions. $$d = 10.58 + 4.6$$ Step 4: Add the values. $$d = 15.18 \text{ m}$$ The stopping distance for $46 \text{ km/hr}$ is $\boxed{\text{15.18 m}}$. To find the stopping distance for $s = 85 \text{ km/hr}$: Step 1: Substitute $s = 85$ into the formula. $$d = \frac{85^2}{200} + \frac{85}{10}$$ Step 2: Calculate $85^2$. $$d = \frac{7225}{200} + \frac{85}{10}$$ Step 3: Perform the divisions. $$d = 36.125 + 8.5$$ Step 4: Add the values. $$d = 44.625 \text{ m}$$ The stopping distance for $85 \text{ km/hr}$ is $\boxed{\text{44.625 m}}$. To find the speed when the stopping distance is $35 \text{ m}$: Step 1: Substitute $d = 35$ into the formula. $$35 = \frac{s^2}{200} + \frac{s}{10}$$ Step 2: Multiply both sides by 200 to eliminate the denominators. $$35 \times 200 = s^2 + \frac{s}{10} \times 200$$ $$7000 = s^2 + 20s$$ Step 3: Rearrange the equation into a standard quadratic form $as^2 + bs + c = 0$. $$s^2 + 20s - 7000 = 0$$ Step 4: Use the quadratic formula $s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a=1$, $b=20$, $c=-7000$. $$s = \frac{-20 \pm \sqrt{20^2 - 4(1)(-7000)}}{2(1)}$$ $$s = \frac{-20 \pm \sqrt{400 + 28000}}{2}$$ $$s = \frac{-20 \pm \sqrt{28400}}{2}$$ Step 5: Calculate the square root. $$\sqrt{28400} \approx 168.5229$$ Step 6: Calculate the two possible values for $s$. $$s_1 = \frac{-20 + 168.5229}{2} = \frac{148.5229}{2} \approx 74.26145$$ $$s_2 = \frac{-20 - 168.5229}{2} = \frac{-188.5229}{2} \approx -94.26145$$ Step 7: Choose the positive value for speed, as speed cannot be negative. $$s \approx 74.26 \text{ km/hr}$$ The speed at which the car is traveling if its stopping distance is $35 \text{ m}$ is $\boxed{\text{74.26 km/hr}}$. Send me the next one 📸