In a flight of 600 km, an aircraft was slowed due to bad weather. Its average speed for the trip was reduced to 200 km/hr and time of flight increased by 30 minutes. Find the original duration of flight.
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Step-by-Step Solution
Step 1: Define variables and formulate equations
We need to find the original duration of the flight. Let's denote the original speed of the aircraft as v km/hr and the original time taken as t hours. The total distance of the flight is given as 600 km.
Step 2: Set up equations based on given information
The fundamental relationship between distance, speed, and time is Distance=Speed×Time. So, for the original flight, we have v⋅t=600. Due to bad weather, the speed was reduced by 200 km/hr, making the new speed v−200. The time of flight increased by 30 minutes, which is 0.5 hours, so the new time is t+0.5. The distance remains 600 km, leading to the second equation: (v−200)(t+0.5)=600.
Step 3: Substitute and simplify the equations
From equation (1), we can express v as v=t600. Substitute this expression for v into equation (2). This will give us an equation solely in terms of t. Expand the equation and simplify.
Step 4: Form a quadratic equation in terms of t
Continue simplifying the equation. Multiply the entire equation by t to eliminate the fraction, and then rearrange the terms to form a standard quadratic equation of the form at2+bt+c=0. Divide by -100 to simplify the coefficients.
Step 5: Solve the quadratic equation for t
We have a quadratic equation 2t2+t−3=0. We can solve this using the quadratic formula t=2a−b±b2−4ac, where a=2, b=1, and c=−3. Calculate the two possible values for t.
Step 6: Determine the valid duration
We get two possible values for t: 1 hour and −1.5 hours. Since time cannot be negative, we discard the negative solution. Therefore, the original duration of the flight is 1 hour.