At midnight, ship B was 90 km due south of ship A. Ship A sailed east at 15 km/hr and ship B sailed north at 20 km/hr. At what time were they closest together?
To determine the time at which ships A and B were closest together, we need to find the point at which their distance is minimal. Let's break the problem down into smaller steps.
Step 1: Determine the initial positions of the ships at midnight:
- Ship A is at an unknown position.
- Ship B is 90 km due south of ship A.
Step 2: Determine the speed at which ships A and B are moving:
- Ship A sails east at 15 km/hr.
- Ship B sails north at 20 km/hr.
Step 3: Define variables for time:
- We'll let t represent the number of hours that have passed since midnight.
Step 4: Determine the positions of the ships at time t:
- Ship A will be at an unknown position, which can be determined using the equation: distance = speed × time.
- Ship B will be 90 km + (20 km/hr × t) north of ship A.
Step 5: Determine the distance between the two ships at time t:
- The horizontal distance between the ships can be calculated as the difference in the x-coordinates of their positions.
- The vertical distance between the ships can be calculated as the difference in the y-coordinates of their positions.
- To find the overall distance between the ships, we can use the Pythagorean theorem: distance = √((horizontal distance)^2 + (vertical distance)^2).
Step 6: Find the time at which the distance between the ships is minimal:
- To find the time at which they are closest together, we need to find the minimum value of the distance function.
- We calculate the derivative of the distance function with respect to time and set it equal to zero.
- We solve the resulting equation to find the value of t that corresponds to the minimum distance.
Following these steps, we can determine the time at which ships A and B were closest together.