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

Options

A. `3 1/2` hours
B. `5 1/2`hours
C. `4 1/2 ` hours
D. `2 1/2` hours

Answer

To solve the problem, we need to set up equations based on the information provided. Let's break it down step by step.

### Step 1: Define Variables
Let the original speed of the aircraft be \( x \) km/h.

### Step 2: Write the Equation for Original Time
The original time taken for the flight can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
So, the original time \( t \) in hours is:
\[ t = \frac{2800}{x} \]

### Step 3: Write the Equation for Reduced Speed
When the speed is reduced by 100 km/h, the new speed becomes:
\[ x - 100 \text{ km/h} \]

### Step 4: Write the Equation for New Time
The new time taken for the flight with the reduced speed is:
\[ t + \frac{1}{2} = \frac{2800}{x - 100} \]
(Note: We add \( \frac{1}{2} \) hour to account for the 30 minutes increase in time.)

### Step 5: Set Up the Equation
Now we can set up the equation based on the two expressions for time:
\[ \frac{2800}{x} + \frac{1}{2} = \frac{2800}{x - 100} \]

### Step 6: Clear the Fractions
To eliminate the fractions, we can multiply through by \( 2x(x - 100) \):
\[ 2x(x - 100) \left( \frac{2800}{x} + \frac{1}{2} \right) = 2x(x - 100) \left( \frac{2800}{x - 100} \right) \]
This simplifies to:
\[ 5600(x - 100) + x(x - 100) = 5600x \]

### Step 7: Expand and Rearrange
Expanding the equation gives:
\[ 5600x - 560000 + x^2 - 100x = 5600x \]
Now, subtract \( 5600x \) from both sides:
\[ x^2 - 100x - 560000 = 0 \]

### Step 8: Solve the Quadratic Equation
We can solve the quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -100, c = -560000 \):
\[ x = \frac{100 \pm \sqrt{(-100)^2 - 4 \cdot 1 \cdot (-560000)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{100 \pm \sqrt{10000 + 2240000}}{2} \]
\[ x = \frac{100 \pm \sqrt{2250000}}{2} \]
\[ x = \frac{100 \pm 1500}{2} \]
This gives us two possible solutions for \( x \):
1. \( x = \frac{1600}{2} = 800 \) km/h
2. \( x = \frac{-1400}{2} = -700 \) km/h (not valid)

### Step 9: Find the Original Duration
Now that we have the original speed \( x = 800 \) km/h, we can find the original duration:
\[ t = \frac{2800}{800} = 3.5 \text{ hours} \]

### Final Answer
The original duration of the flight is **3.5 hours**.

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