To find out how far apart ships A and B are after t hours, we will follow these steps:
1. **Initial Positions**: At noon, let’s set the coordinates based on a Cartesian plane, where:
– Ship A starts at extbf{(-100, 0)} (100 km west of Ship B) and moves east.
– Ship B starts at extbf{(0, 0)} (the origin, where Ship A is currently 100 km to the west) and moves north.
2. **Positions After t Hours**:
– After t hours, Ship A’s position will be:
– x-coordinate: -100 km + (35 km/h * t) = (-100 + 35t) km
– y-coordinate remains 0.
– Therefore, Ship A’s coordinates are extbf{(-100 + 35t, 0)}.
– Ship B’s position after t hours will be:
– x-coordinate remains 0; y-coordinate: 0 km + (25 km/h * t) = (25t) km.
– Thus, Ship B’s coordinates are extbf{(0, 25t)}.
3. **Calculating the Distance**:
The distance d between the two ships after t hours can be obtained using the distance formula:
d = √[(x2 - x1)² + (y2 - y1)²]Substituting the coordinates of the two ships:
d = √[(-100 + 35t - 0)² + (0 - 25t)²]
d = √[(-100 + 35t)² + (-25t)²]Expanding this:
d = √[(10000 - 7000t + 1225t²) + (625t²)]
d = √[10000 - 7000t + 1850t²]Thus, the final expression for the distance between the two ships after t hours is:
d(t) = √[1850t² - 7000t + 10000]This equation gives the distance at any given time t hours after noon.