Two buses leave a city at the same time,one going north and the other east.The north bound bus travels 10km/hr faster than the east bound one.If they are 250km apart after 5hours,What is the speed of each bus?
If one going north and the other east distance will be:
√ { ( 5 x )² + [ 5 ( x + 10 ) ]² } = 250
5 ( x + 10 ) = 5 x + 50
So:
√ [ ( 5 x )² + ( 5 x + 50 )² ] = 250
√ [ 25 x ² + 25 x² + 500 x + 50² ] = 250
√ ( 50 x² + 500 x + 2 500 ) = 250
50 x² + 500 x + 2 500 = 250²
50 x² + 500 x + 2 500 = 62 500
50 x² + 500 x + 2 500 - 62 500 = 0
50 x² + 500 x - 60 000 = 0
The solutions are:
x = - 40 an x = 30
Speed can't be negative, so speed of slower bus:
x = 30 km / h
speed of faster bus = x + 10 = 30 + 10 = 40 km / h
After 5 hrs, the slower went 30 ∙ 5 = 150 km
the faster went 40 ∙ 5 = 200 km
√ [ ( 150)² + ( 200)² ] = √ ( 22 500 + 40 000) = √ 62 500 = 250 km
To find the speed of each bus, let's set up equations based on the given information.
Let's assume the speed of the eastbound bus is x km/hr. Since the northbound bus is traveling 10 km/hr faster, its speed will be x + 10 km/hr.
Now, we know that distance = speed × time.
For the eastbound bus:
Distance covered by the eastbound bus = x km/hr × 5 hours = 5x km.
Similarly, for the northbound bus:
Distance covered by the northbound bus = (x + 10) km/hr × 5 hours = 5(x + 10) km.
Since the buses are traveling perpendicular to each other, we can use the Pythagorean theorem to find the total distance between them. According to the theorem, the sum of the squares of the two shorter sides of a right-angled triangle is equal to the square of the hypotenuse.
In this case, 250 km is the hypotenuse, and the distances traveled by each bus are the two shorter sides. Using the Pythagorean theorem:
(5x)² + (5(x + 10))² = 250²
Simplifying, we get:
25x² + 25(x² + 20x + 100) = 62,500
25x² + 25x² + 500x + 2500 = 62,500
50x² + 500x + 2500 = 62,500
50x² + 500x - 60,000 = 0
Simplifying further, we divide the equation by 50:
x² + 10x - 1200 = 0
We can either factorize or use the quadratic formula to solve this equation. Factoring it, we get:
(x - 30)(x + 40) = 0
So, x = 30 or x = -40
Since we're dealing with speeds, the speed cannot be negative. Therefore, the speed of the eastbound bus is 30 km/hr.
Substituting this value in x + 10, we find the speed of the northbound bus is 30 + 10 = 40 km/hr.
Thus, the speed of the eastbound bus is 30 km/hr, and the speed of the northbound bus is 40 km/hr.