Question

1. the seven dwarves hop into a railroad car and drive 990 km in 9.5 hours. this included a stop of 30 minutes for lunch. they travelled at an average speed of 88 km/h through the railway tunnels and an average speed of 130 km/h in the open air. how much time did they spend travelling through the railway tunnels?

(handwritten calculations and equations around the problem text, including steps like 85×4 + 130×5 = 990, 9.5 - 0.5 = 9h, etc.)

Explanation:

Step1: Find total time

The total time for the journey is 9.5 hours, and they stopped for 0.5 hours for lunch. So the time spent traveling (including through tunnels and open air) is \( 9.5 - 0.5 = 9 \) hours.

Step2: Calculate distance in open air

Their speed in open air is 130 km/h. Let the time in open air be \( x \) hours. But we can also calculate the distance traveled in open air and through tunnels. Wait, actually, let's re - express. The total distance traveled is 990 km. Let the time spent in tunnels be \( y \) hours and in open air be \( x \) hours. We know that \( x + y=9 \) (from step 1, total travel time) and the speed in open air is 130 km/h and in tunnels is 85 km/h. So the distance equation is \( 130x + 85y=990 \). But from \( x = 9 - y \), substitute into the distance equation: \( 130(9 - y)+85y = 990 \).

Step3: Expand and solve for y

Expand \( 130\times9-130y + 85y=990 \), \( 1170-45y = 990 \), \( - 45y=990 - 1170=-180 \), \( y=\frac{- 180}{-45}=4 \) hours. Wait, but let's check the other way. The total distance is 990 km. The distance traveled at 130 km/h and 85 km/h. Let's assume the time in tunnels is \( t \). The total travel time (excluding lunch) is \( 9.5 - 0.5 = 9 \) hours. So distance in open air: \( 130\times(9 - t) \), distance in tunnels: \( 85t \). So \( 130(9 - t)+85t=990 \), \( 1170-130t + 85t=990 \), \( 1170 - 45t=990 \), \( 45t = 1170 - 990 = 180 \), \( t = 4 \) hours. Wait, but the hand - written solution seems to have some miscalculations. Wait, let's re - do it.

Wait, the problem says: "The seven dwarves hop into a railcar and drive 990 km in 9.5 hours. This included a stop of 30 minutes for lunch. They travelled at an average speed of 85 km/h through the railway tunnels and an average speed of 130 km/h in the open air. How much time did they spend travelling through the railway tunnels?"

Total time excluding lunch: \( 9.5-0.5 = 9 \) hours (since 30 minutes = 0.5 hours).

Let \( t \) be the time in tunnels (in hours), then time in open air is \( 9 - t \) hours.

Distance in tunnels: \( 85t \) km, distance in open air: \( 130(9 - t) \) km.

Total distance: \( 85t+130(9 - t)=990 \)

Expand: \( 85t + 1170-130t=990 \)

Combine like terms: \( - 45t=990 - 1170=-180 \)

Solve for \( t \): \( t=\frac{-180}{-45}=4 \) hours.

But looking at the hand - written work, there are some errors. For example, in the hand - written, there is a step \( 85x + 130y = 990 \) and \( x + y=9 \), but maybe the variables were mixed. But the correct calculation gives \( t = 4 \) hours. Wait, but the hand - written answer at the end says "5 hours in railway tunnels & 4 hours in open air" which is wrong. The correct time in tunnels is 4 hours.

Answer:

The time spent traveling through the railway tunnels is 4 hours.