Question
A boat is being pulled into a dock by a rope attached to it, which is passing through a pulley on the dock, positioned 6 meters higher than the boat. If the rope is being pulled in at a rate of 3 meters/sec, at what rate is the boat approaching the dock when it is 8 meters from the dock?
Please enter your answer in decimal format with three significant digits after the decimal point.
Please enter your answer in decimal format with three significant digits after the decimal point.
Answers
Answered by
GPT 3.5
Let's consider the situation using the concept of related rates.
We know that the rope is being pulled in at a rate of 3 meters/sec, which means the length of the rope is changing at that rate. Let's denote the length of the rope as L(t), where t is the time in seconds.
We are interested in finding the rate at which the boat is approaching the dock, which is the rate of change of the distance between the boat and the dock. Let's denote this distance as D(t) and the rate of change as dD/dt.
From the problem statement, we know that the rope is passing through a pulley on the dock, which means the length of the rope and the distance of the boat from the dock are related by the Pythagorean theorem:
L(t)^2 = D(t)^2 + 6^2
Differentiating both sides of this equation with respect to time t, we get:
2L(t) * dL/dt = 2D(t) * dD/dt
We are given dL/dt = 3 meters/sec, and we want to find dD/dt when D(t) = 8 meters.
Substituting the given values and solving for dD/dt:
2(8) * 3 = 2(8) * dD/dt
48 = 16 * dD/dt
dD/dt = 48/16 = 3 meters/sec
Therefore, the boat is approaching the dock at a rate of 3 meters/sec.
We know that the rope is being pulled in at a rate of 3 meters/sec, which means the length of the rope is changing at that rate. Let's denote the length of the rope as L(t), where t is the time in seconds.
We are interested in finding the rate at which the boat is approaching the dock, which is the rate of change of the distance between the boat and the dock. Let's denote this distance as D(t) and the rate of change as dD/dt.
From the problem statement, we know that the rope is passing through a pulley on the dock, which means the length of the rope and the distance of the boat from the dock are related by the Pythagorean theorem:
L(t)^2 = D(t)^2 + 6^2
Differentiating both sides of this equation with respect to time t, we get:
2L(t) * dL/dt = 2D(t) * dD/dt
We are given dL/dt = 3 meters/sec, and we want to find dD/dt when D(t) = 8 meters.
Substituting the given values and solving for dD/dt:
2(8) * 3 = 2(8) * dD/dt
48 = 16 * dD/dt
dD/dt = 48/16 = 3 meters/sec
Therefore, the boat is approaching the dock at a rate of 3 meters/sec.