Ships anchor problem?
A ships anchor is stuck on something 100 fathoms deep. The anchor chain is 500 fathoms long and is being winched in at a constant rate of 2 fathoms per second. Find a formula for the ships speed through the water until the moment that it is directly over the anchor. What is the limiting value.
Thanks for all help!
Comments
First, make a formula for the ship's position (call it "x") as a function of the remaining length of the chain ("L"). It helps to draw a diagram, then use the pythagorean theorem.
x² + (100 fathoms)² = L²
Or:
x = sqrt(L²−(100 fathoms)²)
The ship's speed through the water is dx/dt; so differentiate:
dx/dt = ½(L²−(100 fathoms)²)^(-1/2) × 2L × dL/dt
Simplify:
dx/dt = L/sqrt(L²−(100 fathoms)²) × dL/dt
The problem tells you that L=(500 fathoms - (2 fathoms/sec)(t)); and that dL/dt = -2 fathoms/sec. So plug those into the above, and you have a formula for ship's speed (dx/dt) as a function of time.
> "What is the limiting value."
The speed approaches infinity as L approaches 100 fathoms.
wow u r dumb!