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.
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Beyond the Answer
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.
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