In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km per hour, and by doing so, the time of flight is increased by 30 minutes. Find the original duration of the flight?

The original duration of the flight is 4 hours. By analyzing the speed reduction and increase in time, we formulated a quadratic equation to find the original speed of the aircraft. Finally, we calculated the time based on that speed.
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Answered by Anonymous | 2025-07-04

To solve this problem, we will first define the variables and use the relationship between distance, speed, and time.
Definitions and Equations:

Let the original speed of the aircraft be v km/h.
The distance of the flight is given as 2800 km.
The original time of flight is t hours.

The basic formula that relates speed, distance, and time is: Distance = Speed × Time
From this, we can derive: 2800 = v × t
Now, consider the situation after the speed is reduced by 100 km/h, making the new speed v − 100 km/h. The time is increased by 30 minutes, or 0.5 hours. The equation becomes: 2800 = ( v − 100 ) × ( t + 0.5 )
Solving the Equations:
We have two equations:

v × t = 2800
( v − 100 ) × ( t + 0.5 ) = 2800

Using the first equation, express t in terms of v : t = v 2800 ​
Substitute t into the second equation: ( v − 100 ) × ( v 2800 ​ + 0.5 ) = 2800
Clear the fraction by multiplying through by v : ( v − 100 ) × ( 2800 + 0.5 v ) = 2800 v
Expand the equation: 2800 ( v − 100 ) + 0.5 v ( v − 100 ) = 2800 v
Simplify and collect like terms: 2800 v − 280000 + 0.5 v 2 − 50 v = 2800 v Remove 2800 v from both sides: − 280000 + 0.5 v 2 − 50 v = 0
Multiply through by 2 to clear the fraction: v 2 − 100 v − 560000 = 0
This is a quadratic equation, and it can be solved using the quadratic formula: v = 2 a − b ± b 2 − 4 a c ​ ​ where a = 1 , b = − 100 , and c = − 560000 .
Calculate the discriminant: b 2 − 4 a c = 10 0 2 + 4 × 560000 = 10000 + 2240000 = 2250000
The square root of the discriminant is 1500. Plug values into the quadratic formula: v = 2 100 ± 1500 ​ = { 800 , − 700 }
Since speed cannot be negative, v = 800 km/h.
Calculate the Original Duration:
Using t = v 2800 ​ : t = 800 2800 ​ = 3.5
Therefore, the original duration of the flight was 3.5 hours. This resolves the entire question.

Answered by BenjaminOwenLewis | 2025-07-06