A jet plane traveling at a constant speed goes 1200 miles with the wind, then turns around and travels for 1000 miles against the wind. If the speed of the wind is 50 mph and the total flight took 4 hours, find the speed of the plane in still air.
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Let's denote the speed of the plane in still air as x (in mph). When the plane is flying with the wind, its speed is x + 50 mph, and when it's flying against the wind, its speed is x - 50 mph.
The time it takes to fly 1200 miles with the wind is 1200 / (x + 50) hours, and the time it takes to fly 1000 miles against the wind is 1000 / (x - 50) hours.
According to the problem, the total flight time is 4 hours, so we can write the equation:
1200 / (x + 50) + 1000 / (x - 50) = 4.
To solve this equation, we can multiply all terms by (x + 50)(x - 50) to get rid of the denominators:
1200(x - 50) + 1000(x + 50) = 4(x^2 - 50^2),
1200x - 60000 + 1000x + 50000 = 4x^2 - 10000,
2200x - 10000 = 4x^2 - 10000,
4x^2 - 2200x = 0,
x^2 - 550x = 0,
x(x - 550) = 0.
From this, we get two solutions: x = 0 and x = 550. Since the speed of the plane cannot be zero, the speed of the plane in still air is 550 mph.
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