A car travelling at 50 kilometers per hour crosses a bridge over a river 10 minutes before a boat travelling at 40 kilometers per hour passes under the bridge. The river and the bridge are straight and at right angles to each other. At what rate are the car and the boat separating 10 minutes after the boat passes under the bridge if the height of the bridge is 10 meters.
see the rope and the boat problem below.
Answer please
To solve this problem, we need to find the rate at which the car and the boat are separating after 10 minutes.
First, let's convert the 10 minutes into hours. Since there are 60 minutes in an hour, 10 minutes is equal to 10/60 = 1/6 hour.
Given that the car is traveling at a speed of 50 km/h, and the boat is traveling at a speed of 40 km/h, we can determine the distance covered by each vehicle in 1/6 hour.
For the car:
Distance = Speed × Time
Distance = 50 km/h × (1/6) hour
Distance = 50/6 km
For the boat:
Distance = Speed × Time
Distance = 40 km/h × (1/6) hour
Distance = 40/6 km
Now, we have two right-angled triangles formed by the boat's path under the bridge and the car's path over the bridge. The height of the bridge is given as 10 meters.
Using Pythagoras' theorem, we can find the hypotenuse of the triangles, which represents the distance between the boat and the car.
For the boat's triangle:
Hypotenuse = √((distance)^2 + (height)^2)
Hypotenuse = √((40/6)^2 + 10^2)
Hypotenuse ≈ √(666.67)
For the car's triangle:
Hypotenuse = √((distance)^2 + (height)^2)
Hypotenuse = √((50/6)^2 + 10^2)
Hypotenuse ≈ √(138.89)
Now, to find the rate at which the car and the boat are separating after 10 minutes, we can calculate the difference in distances traveled:
Rate of separation = |(Distance covered by the boat) - (Distance covered by the car)|
Rate of separation = |((40/6) km) - ((50/6) km)|
Rate of separation ≈ |(6.67 km) - (8.33 km)|
Rate of separation ≈ |(-1.66 km)|
Therefore, the car and the boat are separating at a rate of approximately 1.66 km/h after 10 minutes.