At noon, ship A is 100km west of ship B. Ship A is sailing south at 30km/h and ship B is sailing north at 15km/h. How fast is the distance between the ships changing at 4:00pm?
At a time of t hrs,
let the position of ship A be P and let the position of ship B be Q
Join PQ, and complete the large righ-angled triangle
having a base of 100 and a height of 15t + 30t or 45t
(the horizontal distance between them is always 100 km
PQ^2 = 100^2 + (45t)^2
2 PQ d(PQ)/dt = 0 + 2(45t)(45)
d(PQ)/dt = 2025t/D
at 4:00 , t = 4
PQ = √(100^2 + 180^2) = appr 205.913
d(PQ)/dt = 2025(4)/205.913
= 39.34 km/h
Well, isn't that a "ship-stick" situation? Let's "sail" into the problem.
To find the rate at which the distance between the ships is changing, we need to consider their respective velocities. Ship A is sailing south at 30 km/h, while Ship B is sailing north at 15 km/h.
Since they are moving in opposite directions, we can simply add their velocities to find the relative speed between the two ships. The relative speed is 30 km/h + 15 km/h = 45 km/h.
Given that the ships were initially 100 km apart, we can now calculate how fast the distance between them is changing. We're looking for the rate of change at 4:00pm, so let's put on our "time-traveler" hats and calculate.
From noon until 4:00pm, there are 4 hours. Multiplying the relative speed by the duration gives us the change in distance between the ships: 45 km/h * 4 h = 180 km.
Therefore, the distance between the ships is changing at a rate of 180 km over the course of 4 hours, or 45 km/h.
To find the rate at which the distance between the ships is changing, we can use the concept of relative velocity.
Let's first calculate the time difference between noon and 4:00 pm:
4:00 pm - 12:00 pm = 4 hours.
Since ship A is sailing south at 30 km/h and ship B is sailing north at 15 km/h, their velocities are opposite in direction. Thus, we can consider their relative velocity.
The relative velocity of ship A with respect to ship B can be calculated by subtracting their velocities:
Relative velocity = Velocity of A - Velocity of B
= 30 km/h - 15 km/h
= 15 km/h.
Therefore, the distance between the ships will decrease at a rate of 15 km/h.
Hence, at 4:00 pm, the distance between the ships is changing at a rate of 15 km/h.
To find the rate at which the distance between the ships is changing, we can use the concept of relative velocity. Let's break down the problem and solve it step by step:
1. At noon, ship A is 100km west of ship B. This means that the initial distance between the ships is 100km.
2. Ship A is sailing south at a speed of 30km/h. This indicates that the velocity of ship A is pointing in the negative north direction.
3. Ship B is sailing north at a speed of 15km/h. This implies that the velocity of ship B is pointing in the positive north direction.
4. At 4:00pm, we need to find the rate at which the distance between the ships is changing.
To find the rate of change, we can use the concept of relative velocity, which is the difference in velocities of two objects. In this case, we need to calculate the relative velocity of ship A with respect to ship B.
Relative velocity of ship A = Velocity of ship A - Velocity of ship B
Since ship A is moving south and ship B is moving north, their velocities have opposite directions.
Velocity of ship A = -30km/h (negative because it is moving south)
Velocity of ship B = +15km/h (positive because it is moving north)
Relative velocity of ship A = -30km/h - 15km/h = -45km/h
The relative velocity of ship A with respect to ship B is -45km/h. This means that the distance between the ships is decreasing at a rate of 45km/h.
Therefore, at 4:00pm, the distance between the ships is changing at a rate of 45km/h.