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, we can start by letting the average speed of the Hikari train be \( x \) km/h. Consequently, the speed of the Nozomi train will be \( x + 30 \) km/h. The time taken by the Hikari train to travel 400 km is given by \( \frac{400}{x} \) hours, while the time for the Nozomi train is \( \frac{400}{x + 30} \) hours. According to the information provided, the Nozomi train saves 30 minutes (or 0.5 hours) compared to the Hikari train. This information sets up the equation: \[ \frac{400}{x} - \frac{400}{x + 30} = 0.5 \] To eliminate the fractions, we can multiply through by \( x(x + 30) \): \[ 400(x + 30) - 400x = 0.5x(x + 30) \] Simplifying this gives: \[ 12000 = 0.5x^2 + 15x \] Rearranging it leads to the quadratic equation: \[ 0.5x^2 + 15x - 12000 = 0 \] Multiplying the entire equation by 2 to remove the decimal gives: \[ x^2 + 30x - 24000 = 0 \] Now we can solve using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Where \( a = 1, b = 30, c = -24000 \). Plugging and solving gives: \[ x = \frac{-30 \pm \sqrt{30^2 - 4(1)(-24000)}}{2(1)} = \frac{-30 \pm \sqrt{900 + 96000}}{2} = \frac{-30 \pm 310}{2} \] This produces two potential solutions for \( x \): 1. \( \frac{280}{2} = 140 \) (valid since speed cannot be negative) 2. \( \frac{-340}{2} = -170 \) (discarded, as speed can't be negative) Thus, the average speed of the Hikari train is 140 km/h. Therefore, the average speed of the Nozomi train, which is \( x + 30 \), is: \[ 140 + 30 = 170 \text{ km/h} \] So, Renee's research indicates that the average speed of the Nozomi train is **170 km/h**.