A train travels 400 km between Tokyo and Osaka. Renee searches the Internet and finds that the average speed of the Nozomi train is greater than that of t Hikari train by \( 30 \mathrm{~km} / \mathrm{h} \), and the faster train reduces the travelling time by 30 minutes. Find the average speed of the Nozomi train based on Renee's research.

Let the average speed of the Hikari train be \( v \) km/h. Then, the average speed of the Nozomi train is \( v + 30 \) km/h. The time taken for the Hikari train to travel 400 km is: \[ \frac{400}{v} \text{ hours} \] The time taken for the Nozomi train to travel the same distance is: \[ \frac{400}{v + 30} \text{ hours} \] According to the problem, the Nozomi train reduces the travelling time by 30 minutes (or 0.5 hours). Therefore, we have the equation: \[ \frac{400}{v} - \frac{400}{v + 30} = 0.5 \] To solve this equation, first, find a common denominator: \[ \frac{400(v + 30) - 400v}{v(v + 30)} = 0.5 \] This simplifies to: \[ \frac{400 \cdot 30}{v(v + 30)} = 0.5 \] \[ \frac{12000}{v(v + 30)} = 0.5 \] Cross-multiply to eliminate the fraction: \[ 12000 = 0.5v(v + 30) \] \[ 24000 = v(v + 30) \] \[ v^2 + 30v - 24000 = 0 \] Now, we can apply the quadratic formula: \[ v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1, b = 30, c = -24000 \). Calculating the discriminant: \[ b^2 - 4ac = 30^2 - 4 \cdot 1 \cdot (-24000) = 900 + 96000 = 96800 \] Now, solving for \( v \): \[ v = \frac{-30 \pm \sqrt{96800}}{2 \cdot 1} \] Calculating \( \sqrt{96800} \): \[ \sqrt{96800} \approx 311 \] Substituting back: \[ v = \frac{-30 \pm 311}{2} \] Taking the positive root: \[ v \approx \frac{281}{2} = 140.5 \text{ km/h} \] Thus, the average speed of the Nozomi train is: \[ v + 30 \approx 140.5 + 30 = 170.5 \text{ km/h} \] Therefore, the average speed of the Nozomi train is approximately 170.5 km/h.