Step-by-Step Solution

Step-by-Step Solution

Step 1: Set up the quadratic inequality

The braking distance must not exceed $50\text{ m}$: $0.02s^2 + 0.1s \le 50$

Multiply the inequality by $100$: $2s^2 + 10s \le 5000$

Divide by $2$: $s^2 + 5s - 2500 \le 0$

Step 2: Find boundary critical roots

Solve $s^2 + 5s - 2500 = 0$ using the quadratic formula: $s = \frac{-5 \pm \sqrt{5^2 - 4(1)(-2500)}}{2(1)}$

$s = \frac{-5 \pm \sqrt{25 + 10000}}{2} = \frac{-5 \pm \sqrt{10025}}{2}$

Since $\sqrt{10025} \approx 100.125$: $s_1 = \frac{-5 + 100.125}{2} \approx 47.56\text{ km/h}$ $s_2 = \frac{-5 - 100.125}{2} \approx -52.56\text{ km/h}$

Step 3: Determine safe speed interval

For an upward-opening parabola $s^2 + 5s - 2500 \le 0$, the inequality holds between the roots: $-52.56 \le s \le 47.56$

Since vehicle speed is physically non-negative ($s \ge 0$): $0 \le s \le \frac{-5 + \sqrt{10025}}{2} \approx 47.56\text{ km/h}$

Step 4: Check boundary

• At $s = 47.56\text{ km/h}$: $d(47.56) = 0.02(47.56)^2 + 0.1(47.56) \approx 45.24 + 4.76 = 50.0\text{ m} \quad \checkmark$

$\boxed{\text{Safe Speed Range: } 0 \le s \le 47.56\text{ km/h} \quad \left[0, \; \frac{-5+\sqrt{10025}}{2}\right]}$