A boat is pulled in to a dock by a rope with one end attached to the front of the boat and the other end passing through a ring attached to the dock at a point 6 ft higher than the front of the boat. The rope is being pulled through the ring at the rate of 0.2 ft/sec. How fast is the boat approaching the dock when 10 ft of rope is out?
As always, draw a diagram. It is clear that if the length of the rope is z when the boat is x ft from the dock,
z^2 = x^2+36
z dz/dt = x dx/dt
Figure x when z=10, then just plug in your numbers and solve for dx/dt.
To solve this problem, we can use the concept of related rates. Related rates problems involve finding the rate at which one quantity is changing with respect to another related quantity.
Let's define some variables:
- Let x be the distance between the boat and the dock.
- Let y be the length of the rope that is out.
- Let t be the time.
We are given:
- The rate at which the rope is being pulled through the ring, which is 0.2 ft/sec.
We want to find:
- The rate at which the boat is approaching the dock (dx/dt) when 10 ft of rope is out (y = 10 ft).
First, let's establish a relationship between x, y, and the height difference between the dock and the front of the boat.
Since the rope passes through a ring attached to the dock at a point 6 ft higher than the front of the boat, we have:
(x + 6)^2 + y^2 = x^2
Expanding and simplifying, we get:
x^2 + 12x + 36 + y^2 = x^2
12x + 36 + y^2 = 0
12x + y^2 = -36
Next, differentiate both sides of the equation with respect to time:
d(12x)/dt + d(y^2)/dt = d(-36)/dt
12(dx/dt) + 2y(dy/dt) = 0
12(dx/dt) = -2y(dy/dt)
We know that dy/dt = 0.2 ft/sec (given), and we want to find dx/dt when y = 10 ft. Plugging in these values into the equation, we can solve for dx/dt:
12(dx/dt) = -2(10)(0.2)
12(dx/dt) = -4
dx/dt = -4/12
dx/dt = -1/3 ft/sec
Therefore, the boat is approaching the dock at a rate of 1/3 ft/sec when 10 ft of rope is out.
To summarize the steps:
1. Define variables (x, y, t) and establish a relationship between them based on the given information.
2. Differentiate the relationship equation with respect to time (t).
3. Substitute the known values (dy/dt and y) into the derived equation.
4. Solve for dx/dt.