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.

To find the average speed of the Nozomi train based on the information provided, we can set up equations based on the speed and the time saved. Let \( v \) be the average speed of the Hikari train in km/h. Then, the average speed of the Nozomi train would be \( v + 30 \) km/h. The time it takes for the Hikari train to travel 400 km is: \[ \text{Time}_{\text{Hikari}} = \frac{400}{v} \] The time for the Nozomi train is: \[ \text{Time}_{\text{Nozomi}} = \frac{400}{v + 30} \] According to the problem, the Nozomi train reduces the traveling time by 30 minutes (which is 0.5 hours). Thus, we can set up the equation: \[ \frac{400}{v} - \frac{400}{v + 30} = 0.5 \] Now, we multiply through by \( v(v + 30) \) to eliminate the denominators: \[ 400(v + 30) - 400v = 0.5v(v + 30) \] Simplifying this gives: \[ 12000 = 0.5v^2 + 15v \] Multiplying everything by 2 to eliminate the fraction: \[ 24000 = v^2 + 30v \] Rearranging results in: \[ v^2 + 30v - 24000 = 0 \] Now we can use the quadratic formula to solve for \( v \): \[ v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-30 \pm \sqrt{30^2 - 4(1)(-24000)}}{2(1)} \] \[ v = \frac{-30 \pm \sqrt{900 + 96000}}{2} = \frac{-30 \pm \sqrt{96800}}{2} \] Calculating \( \sqrt{96800} \) gives approximately \( 310.87 \): \[ v = \frac{-30 \pm 310.87}{2} \] Taking the positive solution: \[ v = \frac{280.87}{2} \approx 140.43 \text{ km/h} \] Thus, the average speed of the Nozomi train is: \[ v + 30 \approx 140.43 + 30 \approx 170.43 \text{ km/h} \] Therefore, the average speed of the Nozomi train is approximately **170.43 km/h**.