For the first 5 hours of a trip, a plane averaged 120 kilometers per hour. For the remainder of the trip, the plane travelled an average speed of 180 kilometers per hour. If the average speed for the entire trip was 170 kilometers per hour, how many hours long was the entire trip?

In this problem the speed of the plane is constant, so we can use the equations of uniform rectilinear motion, the definition of average speed is the distance traveled between the time taken.

v = d / t

Let's calculate each distance

First part of the trip

v₁ = d₁ / t₁

d₁ = v₁ t₁

d₁ = 120 t₁

Second part of the trip

v₂ = d₂ / t₂

d₂ = v₂ t₂

d₂ = 180 t₂

Total trip

v₃ = d₃ / t₃

d₃ = v₃ t₃

d₃ = 170 t₃

The total travel distance is the sum of each distance and the total time is the initial time of 5 h plus the time of the second part (t2)

d₁ + d₂ = 170 t₃

120 5 + 180 t₂ = 170 (5 + t₂)

Let's solve

600 + 180 t₂ = 850 +170 t₂

t₂ (180 -170) = 850 - 600

10 t₂ = 250

t₂ = 25 h

Therefore, the total travel time is

t all= 5 +25 = 30h