# The first 12 km the tourist walked on foot, and the rest of the way traveled on a racing

The first 12 km the tourist walked on foot, and the rest of the way traveled on a racing motorcycle. How much distance he traveled on a motorcycle, if the driving speed is 160 km / h, the walking speed is 5 km / h and the average speed is 40 km / h.

Sп = 12 km.

Ve = 160 km / h.

Vp = 5 km / h.

Vav = 40 km / h.

Se -?

To find the average speed of movement Vav, it is necessary to divide the entire path traveled by the car S by the time of its movement t along the entire path: Vav = S / t.

S = Sп + Se.

The travel time of the entire path is expressed by the sum: t = tp + te, where tp is the time of walking, tm is the time of movement on a motorcycle.

tp = Sp / Vp.

te = Se / Ve.

t = Sп / Vп + Se / Ve = (Sп * Ve + Se * Vп) / Vп * Ve.

Vav = (Sп + Se) * Vп * Ve / (Sп * Ve + Se * Vп).

Vav * (Sп * Ve + Se * Vп) = Sп * Vп * Ve + Se * Vп * Ve.

Vav * Sp * Ve + Vav * Se * Vp = Sp * Vp * Ve + Se * Vp * Ve.

Vav * Se * Vp – Se * Vp * Ve = Sp * Vp * Ve – Vav * Sp * Ve.

Se = Sп * Ve * (Vп – Vср) / Vп * (Vср – Ve)

Se = 12 km * 160 km / h * (5 km / h – 40 km / h) / 5 km / h * (40 km / h – 160 km / h) = 112 km.

Answer: the tourist covered the distance Se = 112 km on a motorcycle.

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