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Chapter 0: Problem 192

Two planes leave simultaneously from Chicago's O'Hare Airport, one flying due north and the other due east (see figure). The northbound plane is flying 50 miles per hour faster than the eastbound plane. After 3 hours, the planes are 2440 miles apart. Find the speed of each plane.

Short Answer

Expert verified
The eastbound plane is traveling at a speed of approximately 395.46 mph and the northbound plane is traveling at a speed of approximately 445.46 mph.

Step by step solution

01

Identify Unknowns and Assign Variables

Let \( x \) be the speed of the eastbound plane and \( x+50 \) be the speed of the northbound plane. The distance covered by each plane in 3 hours can be calculated as speed multiplied by time (distance = speed * time), so we have \( 3x \) miles for the eastbound plane and \( 3(x+50) \) miles for the northbound plane.
02

Set up an equation using the Pythagorean theorem

The squared distance between the planes (the hypotenuse of the right triangle) is equal to the sum of the squares of the distances each plane has traveled (the legs of the triangle). So, we can write the equation \( (3x)^2 + (3(x+50))^2 = 2440^2 \) . Now we simplify this equation.
03

Simplify the Equation

First expand the brackets: \( 9x^2 + 9(x^2 + 100x + 2500) = 2440^2 \), then combine like terms: \( 18x^2 + 2700x + 22500 = 5967360 \)
04

Solve the Equation

This is a quadratic equation that we can solve for \( x \). After solving, you get \( x \approx 395.46 \) mph for the speed of the eastbound plane and \( x+50 \approx 445.46 \) mph for the speed of the northbound plane.

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