Mary and Sue are both on bikes and leave from point A. Mary travels west at a uniform speed (x mi/hr). Sue travels north at 5mi/hr faster than Mary. After 2 hours, they are 50 miles apart. What is the speed of each girl?
To solve this problem, we can use the Pythagorean theorem to relate the distance between Mary and Sue after 2 hours to their speeds. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's say Mary's speed is x mi/hr. Since she travels west, her distance after 2 hours is 2x miles.
Sue's speed is 5 mi/hr faster than Mary, so her speed is (x + 5) mi/hr. Since she travels north, her distance after 2 hours is 2(x + 5) miles.
Using the Pythagorean theorem, we have:
(2x)^2 + (2(x + 5))^2 = 50^2
Simplifying:
4x^2 + 4(x^2 + 10x + 25) = 2500
4x^2 + 4x^2 + 40x + 100 = 2500
8x^2 + 40x - 2400 = 0
Divide the equation by 8 to simplify:
x^2 + 5x - 300 = 0
This is a quadratic equation that we can solve using factoring, completing the square, or the quadratic formula. Factoring the equation gives us:
(x + 20)(x - 15) = 0
So x = -20 or x = 15.
Since speed cannot be negative, we discard x = -20 and keep x = 15.
Therefore, Mary's speed is 15 mi/hr, and Sue's speed is 20 mi/hr.