A car is traveling on a road that is perpendicular to a railroad track. When the car is 30 meters from the crossing, the car's new collision detector warns the driver that there is a train 50 meters from the car and is heading towards the same crossing. How far is the train from the crossing?
How would I solve this?
your diagram should show a right-angled triangle with sides 30 and hypotenuse 50
using Pythagoras the distance of the train from the crossing is √(50^2 - 30^2) = 40
(You might have recognized the ratio of the 3:4:5 right-angled triangle here)
To solve this problem, we can use the concept of similar triangles.
Let's assume that the distance between the car and the crossing represents the base of a triangle, and the distance between the train and the crossing represents the height of the triangle. Since the car and the train are heading towards the same crossing, the angles between the road, track, and the crossing are all right angles.
According to the problem, the car is 30 meters from the crossing, and the car's collision detector warns the driver that the train is 50 meters away from the car. So, we have the base (30 meters) and the hypotenuse (50 meters) of the triangle formed by the car's position.
Using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can find the height of the triangle (distance between the train and the crossing).
Let's denote the distance between the train and the crossing as 'x.'
Using the Pythagorean theorem:
Hypotenuse^2 = Base^2 + Height^2
50^2 = 30^2 + x^2
Simplifying this equation:
2500 = 900 + x^2
Moving the terms to solve for x:
x^2 = 2500 - 900
x^2 = 1600
Taking the square root of both sides:
x = √1600
x = 40
Therefore, the train is 40 meters away from the crossing.