It took a small plan 1 hour longer to fly 480 miles against the wind than it took the plane to fly the same distance with the wind. If the wind speed was 20 mph, then what is the speed of the plane in calm air?

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To find the speed of the plane in calm air, we can use the concept of relative speed. Let's denote the speed of the plane in calm air as x mph.

When the plane flies with the wind, its effective speed is increased as the wind speed aids its motion. Therefore, the speed of the plane with the wind is (x + 20) mph.

When the plane flies against the wind, its effective speed is decreased as the wind speed opposes its motion. Therefore, the speed of the plane against the wind is (x - 20) mph.

We are given that it took the plane 1 hour longer to fly 480 miles against the wind compared to flying the same distance with the wind.

Using the formula:
Time = Distance / Speed

For the plane flying with the wind:
Time = 480 / (x + 20)

For the plane flying against the wind:
Time = 480 / (x - 20)

Since it took 1 hour longer to fly against the wind, we can set up the following equation:
480 / (x - 20) = 480 / (x + 20) + 1

To solve this equation, we can multiply through by (x - 20) * (x + 20) to eliminate the denominators:

480 * (x + 20) = 480 * (x - 20) + (x - 20) * (x + 20)

Simplifying the equation:
480x + 9600 = 480x - 9600 + x^2 - 400

Combining like terms and moving all terms to one side:
x^2 - 400 = 0

Now we have a quadratic equation. We can solve it by factoring:

(x + 20)(x - 20) = 0

This gives us two possible values for x:
x = -20, x = 20

However, since the speed of the plane cannot be negative, the speed of the plane in calm air is 20 mph.

Therefore, the speed of the plane in calm air is 20 mph.